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2067:{\displaystyle {\begin{alignedat}{8}1&+{\frac {1}{2}}&&+{\frac {1}{3}}&&+{\frac {1}{4}}&&+{\frac {1}{5}}&&+{\frac {1}{6}}&&+{\frac {1}{7}}&&+{\frac {1}{8}}&&+{\frac {1}{9}}&&+\cdots \\{}\geq 1&+{\frac {1}{2}}&&+{\frac {1}{\color {red}{\mathbf {4} }}}&&+{\frac {1}{4}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{8}}&&+{\frac {1}{\color {red}{\mathbf {16} }}}&&+\cdots \\\end{alignedat}}}
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2819:(like the harmonic series) has partial sums that are within a bounded distance of the values of the corresponding integrals. Therefore, the sum converges if and only if the integral over the same range of the same function converges. When this equivalence is used to check the convergence of a sum by replacing it with an easier integral, it is known as the
5483:
2340:{\displaystyle {\begin{aligned}&1+\left({\frac {1}{2}}\right)+\left({\frac {1}{4}}+{\frac {1}{4}}\right)+\left({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}\right)+\left({\frac {1}{16}}+\cdots +{\frac {1}{16}}\right)+\cdots \\{}={}&1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+\cdots .\end{aligned}}}
9305:
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algorithm for sorting a set of items can be analyzed using the harmonic numbers. The algorithm operates by choosing one item as a "pivot", comparing it to all the others, and recursively sorting the two subsets of items whose comparison places them before the pivot and after the pivot. In either its
5244:
harmonic number. The divergence of the harmonic series implies that there is no limit on how far beyond the table the block stack can extend. For stacks with one block per layer, no better solution is possible, but significantly more overhang can be achieved using stacks with more than one block per
4963:
leucas and return, by placing a grain storage depot 5 leucas from the base on the first trip and 12.5 leucas from the base on the second trip. However, Alcuin instead asks a slightly different question, how much grain can be transported a distance of 30 leucas without a final return trip, and either
4024:
is itself even. Therefore, the result is a fraction with an odd numerator and an even denominator, which cannot be an integer. More strongly, any sequence of consecutive integers has a unique member divisible by a greater power of two than all the other sequence members, from which it follows by the
2665:
In the figure to the right, shifting each rectangle to the left by 1 unit, would produce a sequence of rectangles whose boundary lies below the curve rather than above it. This shows that the partial sums of the harmonic series differ from the integral by an amount that is bounded above and below by
1675:
of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit. Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric
5761:
on the right is the evaluation of the convergent series of terms with exponent greater than one. It follows from these manipulations that the sum of reciprocals of primes, on the right hand of this equality, must diverge, for if it converged these steps could be reversed to show that the harmonic
4626:
is the range of distance that the jeep can travel with a single load of fuel. On each trip out and back from the base, the jeep places one more depot, refueling at the other depots along the way, and placing as much fuel as it can in the newly placed depot while still leaving enough for itself to
6357:
analysis of worst-case inputs with a random choice of pivot, all of the items are equally likely to be chosen as the pivot. For such cases, one can compute the probability that two items are ever compared with each other, throughout the recursion, as a function of the number of other items that
5766:, because a finite sum cannot diverge. Although Euler's work is not considered adequately rigorous by the standards of modern mathematics, it can be made rigorous by taking more care with limits and error bounds. Euler's conclusion that the partial sums of reciprocals of primes grow as a
9783:
Knuth writes, of the partial sums of the harmonic series "This sum does not occur very frequently in classical mathematics, and there is no standard notation for it; but in the analysis of algorithms it pops up nearly every time we turn around, and we will consistently use the symbol
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5730:{\displaystyle \ln \prod _{p\in \mathbb {P} }{\frac {1}{1-1/p}}=\sum _{p\in \mathbb {P} }\ln {\frac {1}{1-1/p}}=\sum _{p\in \mathbb {P} }\left({\frac {1}{p}}+{\frac {1}{2p^{2}}}+{\frac {1}{3p^{3}}}+\cdots \right)=\sum _{p\in \mathbb {P} }{\frac {1}{p}}+K.}
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bingo, in which the goal is to obtain all 60 possible numbers of seconds in the times from a sequence of running events. More serious applications of this problem include sampling all variations of a manufactured product for its
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Because each term of the harmonic series is greater than or equal to the corresponding term of the second series (and the terms are all positive), and since the second series diverges, it follows (by the
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4850:: a camel can carry 30 measures of grain and can travel one leuca while eating a single measure, where a leuca is a unit of distance roughly equal to 2.3 kilometres (1.4 mi). The problem has
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Animation of the average-case version of quicksort, with recursive subproblems indicated by shaded arrows and with pivots (red items and blue lines) chosen as the last item in each subproblem
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9300:{\displaystyle {\frac {1}{n}}={\Big (}{\frac {1}{n}}-{\frac {1}{n+1}}{\Big )}+{\Big (}{\frac {1}{n+1}}-{\frac {1}{n+2}}{\Big )}+{\Big (}{\frac {1}{n+2}}-{\frac {1}{n+3}}{\Big )}\cdots }
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value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. Two of the best-known are listed below.
5439:{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i}}=\prod _{p\in \mathbb {P} }\left(1+{\frac {1}{p}}+{\frac {1}{p^{2}}}+\cdots \right)=\prod _{p\in \mathbb {P} }{\frac {1}{1-1/p}},}
4482:(formulated in terms of camels rather than jeeps), but with an incorrect solution. The problem asks how far into the desert a jeep can travel and return, starting from a base with
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1204:
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6581:. The total expected number of comparisons, which controls the total running time of the algorithm, can then be calculated by summing these probabilities over all pairs, giving
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Several common games or recreations involve repeating a random selection from a set of items until all possible choices have been selected; these include the collection of
1154:
9631:{\displaystyle S=\sum _{n=1}^{\infty }{\frac {1}{n}}=\sum _{n=1}^{\infty }\sum _{k=n}^{\infty }{\frac {1}{k(k+1)}}=\sum _{k=1}^{\infty }\sum _{n=1}^{k}{\frac {1}{k(k+1)}}}
5466:
4876:: there are 90 measures of grain, enough to supply three trips. For the standard formulation of the desert-crossing problem, it would be possible for the camel to travel
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from one to infinity that is covered by rectangles) would be less than the area of the union of the rectangles. However, the area under the curve is given by a divergent
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representation. It has been conjectured that every prime number divides the numerators of only a finite subset of the harmonic numbers, but this remains unproven.
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The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value
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1234:
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identical rectangular blocks, one per layer, so that they hang as far as possible over the edge of a table without falling. The top block can be placed with
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loads of fuel, by carrying some of the fuel into the desert and leaving it in depots. The optimal solution involves placing depots spaced at distances
1692:
One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest
2607:
6771:
The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).
1462:
likewise derive from music. Beyond music, harmonic sequences have also had a certain popularity with architects. This was so particularly in the
11415:
7661:
11762:
5936:{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}\left\lfloor {\frac {n}{i}}\right\rfloor \leq {\frac {1}{n}}\sum _{i=1}^{n}{\frac {n}{i}}=H_{n}.}
1688:
There are infinite blue rectangles each with area 1/2, yet their total area is exceeded by that of the grey bars denoting the harmonic series
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11855:
9928:
One might point out that Cauchy's condensation test is merely the extension of Oresme's argument for the divergence of the harmonic series
3405:
216:
2545:
units high, so if the harmonic series converged then the total area of the rectangles would be the sum of the harmonic series. The curve
11171:
7253:{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots ={\frac {\pi }{4}}.}
5043:
of its length extending beyond the next lower block. If it is placed in this way, the next block down needs to be placed with at most
10943:
7101:{\displaystyle {\frac {1}{1}}-{\frac {1}{2}}+\cdots +{\frac {1}{2n-1}}-{\frac {1}{2n}}=H_{2n}-H_{n}=\ln 2-{\frac {1}{2n}}+O(n^{-2})}
11696:
7435:{\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}={\frac {1}{1^{x}}}+{\frac {1}{2^{x}}}+{\frac {1}{3^{x}}}+\cdots ,}
6904:{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }
5810:
8497:{\displaystyle S=1+({\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}})+({\tfrac {1}{5}}+{\tfrac {1}{6}}+{\tfrac {1}{7}})+\cdots }
8750:{\displaystyle S>1+{\tfrac {3}{3}}+{\tfrac {3}{6}}+{\tfrac {3}{9}}+\cdots =1+1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots =1+S}
1280:
11706:
10843:
10389:
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9952:. Classroom Resource Materials Series (2nd ed.). Washington, DC: Mathematical Association of America. pp. 137–138.
8271:
8213:
8141:
2074:
Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent):
10627:
Tsang, Kai-Man (2010). "Recent progress on the
Dirichlet divisor problem and the mean square of the Riemann zeta-function".
2516:. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and
1478:, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
7831:
4821:
harmonic number. The divergence of the harmonic series implies that crossings of any length are possible with enough fuel.
482:
457:
5094:
5046:
1095:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .}
3590:
1656:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots }
2789:{\displaystyle \int _{1}^{N+1}{\frac {1}{x}}\,dx<\sum _{i=1}^{N}{\frac {1}{i}}<\int _{1}^{N}{\frac {1}{x}}\,dx+1.}
11870:
11701:
11461:
11408:
5979:
4964:
strands some camels in the desert or fails to account for the amount of grain consumed by a camel on its return trips.
1493:
in mathematics. However, this achievement fell into obscurity. Additional proofs were published in the 17th century by
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39:
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195:
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8004:
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9776:
9036:
8166:
798:
472:
447:
129:
11027:"The three infinite harmonic series and their sums (with topical reference to the Newton and Leibniz series for
10731:
Gerke, Oke (April 2013). "How much is it going to cost me to complete a collection of football trading cards?".
6762:
1210:. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a
6049:
5973:
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of the top two block is supported and they do not topple. The third block needs to be placed with at most
4986:: blocks aligned according to the harmonic series can overhang the edge of a table by the harmonic numbers
1183:
317:
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6751:
6088:
5135:
of its length extending beyond the next lower block, and so on. In this way, it is possible to place the
4420:
Many well-known mathematical problems have solutions involving the harmonic series and its partial sums.
4340:{\displaystyle \psi (x)={\frac {d}{dx}}\ln {\big (}\Gamma (x){\big )}={\frac {\Gamma '(x)}{\Gamma (x)}}.}
3478:
831:
439:
277:
249:
5480:. The product is divergent, just like the sum, but if it converged one could take logarithms and obtain
4091:
to prove that this set of primes is non-empty. The same argument implies more strongly that, except for
2580:
stays entirely below the upper boundary of the rectangles, so the area under the curve (in the range of
1471:
11757:
10924:
9012:{\displaystyle {\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}+\cdots }
7835:
6547:
2548:
1475:
702:
666:
443:
322:
211:
201:
4355:, the digamma function provides a continuous interpolation of the harmonic numbers, in the sense that
11914:
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7851:
7798:
7618:
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5017:
2519:
2352:) that the harmonic series diverges as well. The same argument proves more strongly that, for every
1467:
1414:
1383:
1352:
466:
6727:{\displaystyle \sum _{i=2}^{n}\sum _{k=0}^{i-2}{\frac {2}{k+2}}=\sum _{i=1}^{n-1}2H_{i}=O(n\log n).}
4764:{\displaystyle {\frac {r}{2n}}+{\frac {r}{2(n-1)}}+{\frac {r}{2(n-2)}}+\cdots ={\frac {r}{2}}H_{n},}
1222:
for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an
302:
10279:
10141:
10075:
9752:{\displaystyle =\sum _{k=1}^{\infty }{\frac {k}{k(k+1)}}=\sum _{k=1}^{\infty }{\frac {1}{k+1}}=S-1}
9088:
Johann
Bernoulli's proof is also by contradiction. It uses a telescopic sum to represent each term
5943:
The operation of rounding each term in the harmonic series to the next smaller integer multiple of
2459:
1242:
1219:
1127:
601:
161:
8136:. MAA Notes. Vol. 77. Washington, DC: Mathematical Association of America. pp. 269–276.
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5449:
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Generalizing this argument, any infinite sum of values of a monotone decreasing positive function
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11664:
10592:
6920:
6916:
6350:
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915:
707:
596:
11228:
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums".
4160:
11504:
11451:
11326:
Schmelzer, Thomas; Baillie, Robert (June 2008). "Summing a curious, slowly convergent series".
10409:
8912:{\displaystyle {\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}}
7577:
6928:
6734:
The divergence of the harmonic series corresponds in this application to the fact that, in the
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2349:
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880:
841:
725:
661:
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Changing the order of summation in the corresponding double series gives, in modern notation
8813:
7571:
4956:{\displaystyle {\tfrac {30}{2}}{\bigl (}{\tfrac {1}{3}}+{\tfrac {1}{2}}+{\tfrac {1}{1}})=27.5}
4412:
This equation can be used to extend the definition to harmonic numbers with rational indices.
11711:
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591:
362:
307:
268:
174:
10527:
Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?".
9075:
8507:
8337:
7280:
6243:
6135:
equally-likely items, the probability of collecting a new item in a single random choice is
5476:
of the terms in the harmonic series, and the right equality uses the standard formula for a
3980:
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11148:
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11060:
11000:
10912:
10894:
10853:
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10568:
10529:
10476:
10375:
10217:
10188:
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10019:
9967:
9834:
9787:
9433:{\displaystyle ={\frac {1}{n(n+1)}}+{\frac {1}{(n+1)(n+2)}}+{\frac {1}{(n+2)(n+3)}}\cdots }
8319:
8281:
7979:
7719:
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910:
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312:
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4028:
Another proof that the harmonic numbers are not integers observes that the denominator of
1159:
8:
11802:
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7484:
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6521:
6168:
6138:
6070:
Graph of number of items versus the expected number of trials needed to collect all items
5763:
5473:
4853:
4627:
return to the previous depots and the base. Therefore, the total distance reached on the
4441:
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875:
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403:
259:
142:
137:
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10992:
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1505:
for finding the proof, and it was later included in Johann
Bernoulli's collected works.
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11643:
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11600:
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11004:
10988:
10805:
10788:
Luko, Stephen N. (March 2009). "The "coupon collector's problem" and quality control".
10748:
10713:
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It is possible to prove that the harmonic series diverges by comparing its sum with an
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773:
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8613:{\displaystyle {\tfrac {1}{x-1}}+{\tfrac {1}{x}}+{\tfrac {1}{x+1}}>{\tfrac {3}{x}}}
8258:. Undergraduate Texts in Mathematics (3rd ed.). New York: Springer. p. 182.
11832:
11633:
11605:
11559:
11549:
11529:
11514:
11376:
11289:
Baillie, Robert (May 1979). "Sums of reciprocals of integers missing a given digit".
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on a different series, marked the first appearance of infinite series other than the
1177:
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556:
434:
387:
244:
239:
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11008:
7110:
Using alternating signs with only odd unit fractions produces a related series, the
11817:
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11554:
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11341:
11337:
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11300:
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9999:
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9049:
9045:
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to be the only values other than negative integers where the function can be zero.
6238:
shows that the total expected number of random choices needed to collect all items
5814:
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5477:
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4207:
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on how many random trials are needed to provide a complete range of responses, the
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682:
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Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
6048:). Bounding the final error term more precisely remains an open problem, known as
11595:
11524:
11263:
11144:
11104:
11078:
11022:
10996:
10890:
10871:. Undergraduate Texts in Mathematics. New York: Springer-Verlag. pp. 80–82.
10849:
10662:
10609:
10590:
Pollack, Paul (2015). "Euler and the partial sums of the prime harmonic series".
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10472:
10213:
10168:
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8787:
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8277:
8104:. In fact, when all the terms containing any particular string of digits (in any
7839:
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6084:
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causes this average to differ from the harmonic numbers by a small constant, and
4599:{\displaystyle {\tfrac {r}{2n}},{\tfrac {r}{2(n-1)}},{\tfrac {r}{2(n-2)}},\dots }
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11486:
11471:
11251:
10963:
10920:
10823:
10498:
10381:
10371:
9943:
8333:
8251:
8083:
7477:
6353:(with the assumption that all input permutations are equally likely) or in its
6163:
and the expected number of random choices needed until a new item is collected
6045:
5260:
5088:
4223:
1494:
803:
611:
378:
11140:
10876:
10801:
10770:
10648:
10552:
10244:
10231:
Sofo, Anthony; Srivastava, H. M. (2015). "A family of shifted harmonic sums".
10154:
10088:
8346:
New arithmetic quadrature (i.e., integration), or On the addition of fractions
8263:
7606:
6767:
4472:
or desert-crossing problem is included in a 9th-century problem collection by
11908:
11792:
11466:
10984:
10835:
10468:
10446:
10442:
10434:
10363:
8229:
7827:
7567:
6354:
5782:
5738:
5268:
4348:
2455:
1664:
1482:
1447:
1215:
984:
783:
547:
297:
254:
9876:
8164:
Kullman, David E. (May 2001). "What's harmonic about the harmonic series?".
8134:
Mathematical Time
Capsules: Historical Modules for the Mathematics Classroom
5770:
of the number of terms has been confirmed by later mathematicians as one of
5468:
denotes the set of prime numbers. The left equality comes from applying the
11797:
11539:
11481:
11073:
11026:
10367:
10313:
See problem 52: De homine patrefamilias – A lord of the manor, pp. 124–125.
10073:(November 2012). "96.53 Partial sums of series that cannot be an integer".
9772:
8820:
Theory of inference, posthumous work. With the
Treatise on infinite series…
6092:
6075:
6066:
5272:
4469:
4429:
4056:
3736:
1693:
1540:
1246:
537:
282:
10273:(March 1992). "Problems to sharpen the young: An annotated translation of
9893:
9034:
Dunham, William (January 1987). "The
Bernoullis and the harmonic series".
8370:
denote the sum of the series. Group the terms of the series in triplets:
6544:
items is equally likely to be chosen first, this happens with probability
11544:
11491:
10766:
10438:
9985:
8105:
7510:
6739:
6336:
5762:
series also converges, which it does not. An immediate corollary is that
4978:
3365:
2662:
Because this integral does not converge, the sum cannot converge either.
1672:
972:
900:
11349:
10964:"On Riemann's rearrangement theorem for the alternating harmonic series"
10657:
10163:
10096:
9923:
9858:
a "harmonic number" because is customarily called the harmonic series."
8815:
Ars conjectandi, opus posthumum. Accedit
Tractatus de seriebus infinitis
8128:
Rice, Adrian (2011). "The harmonic series: A primer". In
Jardine, Dick;
1273:
11393:
11312:
11259:
11200:
10916:
10744:
10709:
10349:
10323:
10300:
10209:
10011:
9057:
8187:
5264:
4464:, showing the amount of fuel in each depot and in the jeep at each step
1345:
1254:
646:
570:
292:
287:
191:
8792:
Propositiones arithmeticae de seriebus infinitis earumque summa finita
5785:
closely related to the harmonic series concerns the average number of
2447:{\displaystyle \sum _{n=1}^{2^{k}}{\frac {1}{n}}\geq 1+{\frac {k}{2}}}
11476:
11384:
10605:
10450:
8796:
Arithmetical propositions about infinite series and their finite sums
6345:
6331:
4352:
2474:
Rectangles with area given by the harmonic series, and the hyperbola
1262:
575:
565:
11192:
10867:
Isaac, Richard (1995). "8.4 The coupon collector's problem solved".
10701:
10292:
9086:. Lausanne & Basel: Marc-Michel Bousquet & Co. vol. 4, p. 8.
8179:
3550:{\displaystyle H_{n}=\ln n+\gamma +{\frac {1}{2n}}-\varepsilon _{n}}
11424:
11242:
10543:
9890:(2). American Mathematical Association of Two-Year Colleges: 31–43.
1337:
1223:
641:
383:
340:
29:
1481:
The divergence of the harmonic series was first proven in 1350 by
9778:
The Art of
Computer Programming, Volume I: Fundamental Algorithms
6079:
5786:
5087:
of its length extending beyond the next lower block, so that the
4434:
4025:
same argument that no two harmonic numbers differ by an integer.
2357:
1463:
1233:
Applications of the harmonic series and its partial sums include
3477:, as can be seen from the integral test. More precisely, by the
2470:
6418:
other items, then the algorithm will make a comparison between
4473:
3864:
can be rewritten as a sum of fractions with equal denominators
2655:{\displaystyle \int _{1}^{\infty }{\frac {1}{x}}\,dx=\infty .}
10929:(3rd ed.). MIT Press and McGraw-Hill. pp. 170–190.
8924:[16. The sum of an infinite series of harmonic progression,
5976:
showed more precisely that the average number of divisors is
1253:
on how far over the edge of a table a stack of blocks can be
1446:. Every term of the harmonic series after the first is the
1336:
The name of the harmonic series derives from the concept of
10688:
Maunsell, F. G. (October 1938). "A problem in cartophily".
10505:[Various observations concerning infinite series].
8342:
Novae quadraturae arithmeticae, seu De additione fractionum
10949:
10907:
10433:
5472:
to the product and recognizing the resulting terms as the
1235:
Euler's proof that there are infinitely many prime numbers
10830:. Cambridge University Press, Cambridge. pp. 64–68.
10326:(May 1970). "The jeep once more or jeeper by the dozen".
11374:
7745:
independent and identically distributed random variables
7709:{\displaystyle \sum _{n=1}^{\infty }{\frac {s_{n}}{n}},}
1326:{\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},\dots }
10952:, Section 8.1, "Lower bounds for sorting", pp. 191–193.
4202:
10771:"The coupon collector's problem (with Geoff Marshall)"
9096:
8992:
8977:
8962:
8947:
8932:
8898:
8883:
8868:
8853:
8838:
8830:
XVI. Summa serei infinita harmonicè progressionalium,
8718:
8703:
8670:
8655:
8640:
8599:
8576:
8561:
8538:
8474:
8459:
8444:
8423:
8408:
8393:
8208:. University of Chicago Press. pp. 11–12, 37–51.
8009:
7856:
7803:
7623:
7513:. Other important values of the zeta function include
6934:
Explicitly, the asymptotic expansion of the series is
6552:
5951:
5255:
Divergence of the sum of the reciprocals of the primes
5163:
5114:
5099:
5066:
5051:
5022:
4933:
4918:
4903:
4884:
4562:
4530:
4510:
3930:{\displaystyle H_{n}=\sum _{i=1}^{n}{\tfrac {M/i}{M}}}
3906:
2559:
2524:
1419:
1388:
1357:
1306:
1291:
11773:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
11763:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
11113:
11033:
10125:
9837:
9817:
9790:
9643:
9446:
9312:
9123:
9094:
8930:
8836:
8763:
8626:
8536:
8510:
8376:
8356:
8043:
8007:
7982:
7955:
7932:
7908:
7885:
7854:
7801:
7776:
7753:
7722:
7664:
7621:
7580:
7520:
7487:
7448:
7309:
7283:
7122:
6942:
6781:
6587:
6550:
6524:
6504:
6484:
6464:
6444:
6424:
6404:
6384:
6364:
6306:
6278:
6246:
6221:
6201:
6171:
6141:
6121:
6101:
5982:
5949:
5823:
5795:
5747:
5486:
5452:
5281:
5228:
5200:
5161:
5141:
5097:
5049:
5020:
5000:
4882:
4856:
4830:
4805:
4777:
4653:
4633:
4612:
4508:
4488:
4444:
4362:
4231:
4163:
4130:
4097:
4065:
4034:
4010:
3983:
3944:
3870:
3843:
3821:
3795:
3773:
3744:
3714:
3679:
3650:
3593:
3563:
3487:
3408:
3380:
3350:
2867:
2845:
2804:
2672:
2610:
2586:
2551:
2522:
2480:
2386:
2365:
2080:
1702:
1557:
1518:
1417:
1386:
1355:
1283:
1186:
1162:
1130:
1110:
993:
42:
10362:
9875:
Kifowit, Steven J.; Stamps, Terra A. (Spring 2006).
8919:&c. est infinita. Id primus deprehendit Frater:…
7468:
would be the harmonic series. It can be extended by
5128:{\displaystyle {\tfrac {1}{2}}\cdot {\tfrac {1}{3}}}
5080:{\displaystyle {\tfrac {1}{2}}\cdot {\tfrac {1}{2}}}
3459:{\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}.}
8206:
Architecture and Geometry in the Age of the Baroque
4824:For instance, for Alcuin's version of the problem,
3637:{\displaystyle 0\leq \varepsilon _{n}\leq 1/8n^{2}}
1508:The partial sums of the harmonic series were named
1466:period, when architects used them to establish the
1214:. Its divergence was proven in the 14th century by
11227:
11119:
11039:
10526:
10131:
9850:
9823:
9803:
9751:
9630:
9432:
9299:
9109:
9011:
8911:
8769:
8749:
8612:
8522:
8496:
8362:
8304:. Washington, DC: Joseph Henry Press. p. 10.
8065:
8028:
7991:
7964:
7938:
7914:
7894:
7869:
7816:
7785:
7762:
7735:
7708:
7636:
7613:, and the "critical line" of complex numbers with
7595:
7556:
7499:
7460:
7434:
7295:
7252:
7100:
6903:
6738:used for quicksort, it is not possible to sort in
6726:
6573:
6536:
6510:
6490:
6470:
6450:
6430:
6410:
6390:
6370:
6358:separate them in the final sorted order. If items
6312:
6291:
6262:
6227:
6207:
6185:
6155:
6127:
6115:items remaining to be collected out of a total of
6107:
6036:
5964:
5935:
5801:
5753:
5729:
5460:
5438:
5234:
5213:
5186:
5147:
5127:
5079:
5035:
5006:
4955:
4868:
4842:
4811:
4790:
4763:
4639:
4618:
4598:
4494:
4456:
4402:
4339:
4182:
4149:
4116:
4079:
4047:
4016:
3995:
3965:
3929:
3856:
3827:
3801:
3779:
3757:
3727:
3698:
3656:
3636:
3575:
3549:
3458:
3393:
3356:
2880:
2851:
2810:
2788:
2654:
2592:
2572:
2537:
2508:through the upper left corners of these rectangles
2500:
2446:
2371:
2339:
2066:
1655:
1531:
1432:
1401:
1370:
1325:
1198:
1168:
1148:
1116:
1094:
114:
9289:
9243:
9233:
9187:
9177:
9139:
9019:, is infinite. My brother first discovered this…]
6458:only when, as the recursion progresses, it picks
6037:{\displaystyle \ln n+2\gamma -1+O(1/{\sqrt {n}})}
4347:Just as the gamma function provides a continuous
1485:. Oresme's work, and the contemporaneous work of
11906:
11325:
10507:Commentarii Academiae Scientiarum Petropolitanae
10268:
9894:More proofs of divergence of the harmonic series
115:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
10826:; Karoński, Michał (2016). "4.1 Connectivity".
9988:(1971). "Partial sums of the harmonic series".
9781:(1st ed.). Addison-Wesley. pp. 73–78.
5248:
10822:
10230:
10046:. Princeton University Press. pp. 21–25.
9877:"The harmonic series diverges again and again"
7924:and decreases to near-zero for values greater
6763:Riemann series theorem § Changing the sum
6756:
4606:from the starting point and each other, where
1671:: as more terms of the series are included in
1450:of the neighboring terms, so the terms form a
11409:
10503:"Variae observationes circa series infinitas"
9874:
6095:. In situations of this form, once there are
4897:
4289:
4270:
3735:is not an integer is to consider the highest
3672:No harmonic numbers are integers, except for
952:
11856:Hypergeometric function of a matrix argument
8029:{\displaystyle {\tfrac {1}{8}}-\varepsilon }
1546:
18:Divergent sum of all positive unit fractions
11712:1 + 1/2 + 1/3 + ... (Riemann zeta function)
9938:
9936:
8822:]. Basel: Thurneysen. pp. 250–251.
8077:
6498:as a pivot before picking any of the other
1663:in which the terms are all of the positive
1551:The harmonic series is the infinite series
1348:of the overtones of a vibrating string are
1277:A wave and its harmonics, with wavelengths
11416:
11402:
11107:(2010). "The classical theory of zeta and
10186:Ross, Bertram (1978). "The psi function".
9980:
9870:
9868:
9866:
9864:
8294:
7974:Intermediate between these ranges, at the
6518:items between them. Because each of these
959:
945:
11768:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
11319:
11241:
11169:
11163:
11072:
10656:
10542:
10522:
10520:
10264:
10262:
10162:
9070:
9029:
9027:
8808:
8786:
8350:Mengoli's proof is by contradiction: Let
8348:] (in Latin). Bologna: Giacomo Monti.
8288:
8250:
8228:
8159:
8157:
8155:
8153:
7653:
5702:
5605:
5552:
5505:
5454:
5401:
5328:
2773:
2704:
2636:
1124:terms of the series sum to approximately
75:
11423:
11103:
11097:
10961:
10955:
10687:
10681:
10427:
10403:
10401:
10065:
10063:
10033:
10031:
10029:
9942:
9933:
9906:Roy, Ranjan (December 2007). "Review of
9767:
9765:
8244:
8066:{\displaystyle \varepsilon <10^{-42}}
7262:
6766:
6335:
6065:
5774:, and can be seen as a precursor to the
5737:Here, each logarithm is replaced by its
4977:
4433:
4403:{\displaystyle \psi (n)=H_{n-1}-\gamma }
4201:
3364:terms of the harmonic series produces a
2469:
1683:
1272:
11288:
11282:
10759:
10589:
10583:
9861:
8332:
8163:
6325:
5764:there are infinitely many prime numbers
483:Differentiating under the integral sign
11907:
11221:
10816:
10765:
10517:
10356:
10259:
10139:-adic valuation of harmonic numbers".
9974:
9064:
9033:
9024:
8326:
8238:Questions concerning Euclid's Geometry
8200:
8150:
8123:
8121:
5155:blocks in such a way that they extend
2666:the unit area of the first rectangle:
2462:is a generalization of this argument.
11397:
11375:
11021:
11015:
10866:
10860:
10730:
10724:
10626:
10620:
10497:
10491:
10407:
10398:
10118:
10112:
10069:
10060:
10037:
10026:
9771:
9762:
8802:
8780:
8757:, which is impossible for any finite
8234:Quaestiones super Geometriam Euclidis
8222:
8194:
8108:) are removed, the series converges.
6055:
5267:, the harmonic series is equal to an
3937:in which only one of the numerators,
3576:{\displaystyle \gamma \approx 0.5772}
3473:These numbers grow very slowly, with
10901:
10787:
10781:
10322:
10316:
10185:
10179:
10109:See in particular Theorem 1, p. 516.
8127:
7557:{\displaystyle \zeta (2)=\pi ^{2}/6}
5789:of the numbers in a range from 1 to
5187:{\displaystyle {\tfrac {1}{2}}H_{n}}
4423:
2861:Partial sum of the harmonic series,
2454:This is the original proof given by
1199:{\displaystyle \gamma \approx 0.577}
11733:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
9949:A Radical Approach to Real Analysis
9908:A Radical Approach to Real Analysis
9905:
9899:
9831:stands for "harmonic", and we call
8118:
8036:for a nonzero but very small value
13:
10410:"Problem 52: Overhanging dominoes"
9775:(1968). "1.2.7 Harmonic numbers".
9714:
9663:
9575:
9524:
9503:
9469:
8777:. Therefore, the series diverges.
7681:
7509:where the extended function has a
7341:
7139:
6798:
5298:
4967:
4319:
4301:
4275:
3975:is odd and the rest are even, and
2646:
2621:
1679:
1574:
1010:
24:Part of a series of articles about
14:
11926:
11851:Generalized hypergeometric series
11368:
11329:The American Mathematical Monthly
11292:The American Mathematical Monthly
11180:The American Mathematical Monthly
10972:The American Mathematical Monthly
10923:(2009) . "Chapter 7: Quicksort".
10456:The American Mathematical Monthly
10329:The American Mathematical Monthly
10275:Propositiones ad acuendos juvenes
10044:Gamma: Exploring Euler's Constant
9991:The American Mathematical Monthly
6745:
6574:{\displaystyle {\tfrac {2}{k+2}}}
4479:Propositiones ad Acuendos Juvenes
4438:Solution to the jeep problem for
2573:{\displaystyle y={\tfrac {1}{x}}}
2038:
1996:
1972:
1948:
1906:
1512:, and given their usual notation
1501:. Bernoulli credited his brother
11889:
11888:
11861:Lauricella hypergeometric series
11579:
11052:Proceedings of the Royal Society
10374:(1989). "6.3 Harmonic numbers".
10040:"Chapter 2: The harmonic series"
9892:See also unpublished addendum, "
5271:in which each term comes from a
5194:lengths beyond the table, where
4197:
4190:, no harmonic number can have a
2465:
2041:
1999:
1975:
1951:
1909:
11871:Riemann's differential equation
10962:Freniche, Francisco J. (2010).
10224:
9110:{\displaystyle {\tfrac {1}{n}}}
9037:The College Mathematics Journal
8167:The College Mathematics Journal
7870:{\displaystyle {\tfrac {1}{4}}}
7832:Kolmogorov three-series theorem
7817:{\displaystyle {\tfrac {1}{2}}}
7637:{\displaystyle {\tfrac {1}{2}}}
5965:{\displaystyle {\tfrac {1}{n}}}
5036:{\displaystyle {\tfrac {1}{2}}}
4415:
3667:
2826:
2538:{\displaystyle {\tfrac {1}{n}}}
1433:{\displaystyle {\tfrac {1}{4}}}
1402:{\displaystyle {\tfrac {1}{3}}}
1371:{\displaystyle {\tfrac {1}{2}}}
983:formed by summing all positive
11342:10.1080/00029890.2008.11920559
11305:10.1080/00029890.1979.11994810
10342:10.1080/00029890.1970.11992525
10202:10.1080/0025570X.1978.11976704
10004:10.1080/00029890.1971.11992881
9689:
9677:
9622:
9610:
9550:
9538:
9421:
9409:
9406:
9394:
9379:
9367:
9364:
9352:
9337:
9325:
9050:10.1080/07468342.1987.11973001
8485:
8440:
8434:
8389:
7830:, as can be seen by using the
7658:The random harmonic series is
7590:
7584:
7530:
7524:
7319:
7313:
7157:
7147:
7095:
7079:
6816:
6806:
6718:
6703:
6031:
6013:
5974:Peter Gustav Lejeune Dirichlet
4944:
4723:
4711:
4693:
4681:
4583:
4571:
4551:
4539:
4372:
4366:
4328:
4322:
4314:
4308:
4284:
4278:
4241:
4235:
3468:
109:
103:
94:
88:
72:
66:
1:
11866:Modular hypergeometric series
11707:1/4 + 1/16 + 1/64 + 1/256 + ⋯
11170:Schmuland, Byron (May 2003).
10828:Introduction to Random Graphs
10119:Sanna, Carlo (2016). "On the
8338:"Praefatio [Preface]"
8111:
7838:. The sum of the series is a
7836:Kolmogorov maximal inequality
2821:integral test for convergence
1228:integral test for convergence
1149:{\displaystyle \ln n+\gamma }
409:Integral of inverse functions
11129:Milan Journal of Mathematics
10869:The Pleasures of Probability
10453:(2009). "Maximum overhang".
7844:probability density function
5461:{\displaystyle \mathbb {P} }
5249:Counting primes and divisors
7:
11876:Theta hypergeometric series
11230:Comptes Rendus Mathématique
8256:Mathematics and its History
8001:the probability density is
6913:alternating harmonic series
6757:Alternating harmonic series
6752:List of sums of reciprocals
6736:comparison model of sorting
6195:Summing over all values of
6050:Dirichlet's divisor problem
4994:, one must place a pile of
832:Calculus on Euclidean space
250:Logarithmic differentiation
10:
11931:
11758:Infinite arithmetic series
11702:1/2 + 1/4 + 1/8 + 1/16 + ⋯
11697:1/2 − 1/4 + 1/8 − 1/16 + ⋯
11252:10.1016/j.crma.2018.11.007
10926:Introduction to Algorithms
8081:
7834:or of the closely related
7266:
6760:
6749:
6329:
6062:Coupon collector's problem
6059:
5252:
4971:
4427:
4183:{\displaystyle H_{6}=2.45}
4055:must be divisible by all
3332:
3310:
3288:
3266:
3244:
3222:
3200:
3178:
3156:
3134:
3112:
3090:
3068:
3046:
3024:
3002:
2980:
2958:
2936:
2914:
2830:
1268:
1239:coupon collector's problem
11884:
11841:
11785:
11720:
11689:
11682:
11652:
11621:
11614:
11588:
11577:
11500:
11444:
11435:
11141:10.1007/s00032-010-0121-8
10877:10.1007/978-1-4612-0819-8
10802:10.1080/08982110802642555
10649:10.1007/s11425-010-4068-6
10561:10.4169/math.mag.90.5.355
10553:10.4169/math.mag.90.5.355
10245:10.1007/s11139-014-9600-9
10155:10.1016/j.jnt.2016.02.020
10089:10.1017/S0025557200005167
8264:10.1007/978-1-4419-6053-5
7747:that take the two values
7596:{\displaystyle \zeta (3)}
7303:by the convergent series
4150:{\displaystyle H_{2}=1.5}
3585:Euler–Mascheroni constant
2905:
2891:
2860:
2839:
1547:Definition and divergence
1218:using a precursor to the
1208:Euler–Mascheroni constant
566:Summand limit (term test)
11172:"Random harmonic series"
10993:10.4169/000298910x485969
10985:10.4169/000298910X485969
10836:10.1017/CBO9781316339831
10690:The Mathematical Gazette
10469:10.4169/000298909X474855
10280:The Mathematical Gazette
10142:Journal of Number Theory
10076:The Mathematical Gazette
8798:]. Basel: J. Conrad.
8078:Depleted harmonic series
6917:conditionally convergent
2460:Cauchy condensation test
1220:Cauchy condensation test
245:Implicit differentiation
235:Differentiation notation
162:Inverse function theorem
11589:Properties of sequences
10593:Elemente der Mathematik
9910:by David M. Bressoud".
8825:From p. 250, prop. 16:
7828:with probability 1
6921:alternating series test
6351:average-case complexity
4117:{\displaystyle H_{1}=1}
3966:{\displaystyle M/2^{k}}
3699:{\displaystyle H_{1}=1}
3479:Euler–Maclaurin formula
2892:expressed as a fraction
708:Helmholtz decomposition
11452:Arithmetic progression
11121:
11074:10.1098/rspa.1943.0026
11041:
10133:
10038:Havil, Julian (2003).
9852:
9825:
9805:
9753:
9718:
9667:
9632:
9600:
9579:
9528:
9507:
9473:
9434:
9301:
9111:
9013:
8913:
8771:
8751:
8614:
8524:
8523:{\displaystyle x>1}
8498:
8364:
8067:
8030:
7993:
7966:
7940:
7916:
7896:
7871:
7818:
7787:
7764:
7737:
7710:
7685:
7654:Random harmonic series
7638:
7597:
7558:
7501:
7462:
7436:
7345:
7297:
7296:{\displaystyle x>1}
7254:
7143:
7102:
6929:natural logarithm of 2
6905:
6802:
6772:
6728:
6683:
6635:
6608:
6575:
6538:
6512:
6492:
6472:
6452:
6432:
6412:
6392:
6372:
6341:
6314:
6293:
6264:
6263:{\displaystyle nH_{n}}
6229:
6209:
6187:
6157:
6129:
6109:
6078:and the completion of
6071:
6038:
5966:
5937:
5906:
5854:
5803:
5755:
5731:
5462:
5440:
5302:
5236:
5215:
5188:
5149:
5129:
5081:
5037:
5008:
4992:block-stacking problem
4987:
4984:block-stacking problem
4974:Block-stacking problem
4957:
4870:
4844:
4813:
4792:
4765:
4641:
4620:
4600:
4496:
4465:
4458:
4404:
4341:
4220:logarithmic derivative
4211:
4210:on the complex numbers
4184:
4151:
4118:
4081:
4049:
4018:
3997:
3996:{\displaystyle n>1}
3967:
3931:
3904:
3858:
3829:
3803:
3781:
3759:
3729:
3708:One way to prove that
3700:
3658:
3644:which approaches 0 as
3638:
3577:
3551:
3460:
3442:
3395:
3358:
2882:
2853:
2812:
2790:
2734:
2656:
2594:
2574:
2539:
2509:
2502:
2448:
2414:
2373:
2341:
2068:
1689:
1657:
1578:
1533:
1444:fundamental wavelength
1442:etc., of the string's
1434:
1403:
1372:
1333:
1327:
1251:block-stacking problem
1237:, the analysis of the
1200:
1170:
1150:
1118:
1096:
1014:
842:Limit of distributions
662:Directional derivative
318:Faà di Bruno's formula
116:
11843:Hypergeometric series
11457:Geometric progression
11122:
11042:
10913:Leiserson, Charles E.
10769:(February 12, 2022).
10417:Pi Mu Epsilon Journal
10408:Sharp, R. T. (1954).
10233:The Ramanujan Journal
10134:
9853:
9851:{\displaystyle H_{n}}
9826:
9806:
9804:{\displaystyle H_{n}}
9754:
9698:
9647:
9633:
9580:
9559:
9508:
9487:
9453:
9435:
9302:
9112:
9014:
8914:
8772:
8752:
8615:
8525:
8499:
8365:
8068:
8031:
7994:
7992:{\displaystyle \pm 2}
7967:
7941:
7917:
7897:
7872:
7819:
7788:
7765:
7738:
7736:{\displaystyle s_{n}}
7711:
7665:
7639:
7598:
7559:
7502:
7470:analytic continuation
7463:
7437:
7325:
7298:
7275:Riemann zeta function
7269:Riemann zeta function
7263:Riemann zeta function
7255:
7123:
7103:
6925:absolutely convergent
6906:
6782:
6770:
6729:
6657:
6609:
6588:
6576:
6539:
6513:
6493:
6473:
6453:
6433:
6413:
6393:
6373:
6339:
6315:
6294:
6292:{\displaystyle H_{n}}
6265:
6230:
6210:
6188:
6158:
6130:
6110:
6069:
6039:
5967:
5938:
5886:
5834:
5804:
5756:
5732:
5463:
5441:
5282:
5237:
5216:
5214:{\displaystyle H_{n}}
5189:
5150:
5130:
5082:
5038:
5009:
4981:
4958:
4871:
4845:
4814:
4793:
4791:{\displaystyle H_{n}}
4766:
4642:
4621:
4601:
4497:
4459:
4437:
4405:
4342:
4205:
4185:
4152:
4119:
4082:
4050:
4048:{\displaystyle H_{n}}
4019:
3998:
3968:
3932:
3884:
3859:
3857:{\displaystyle H_{k}}
3830:
3811:least common multiple
3804:
3782:
3760:
3758:{\displaystyle 2^{k}}
3730:
3728:{\displaystyle H_{n}}
3701:
3659:
3639:
3578:
3552:
3461:
3422:
3396:
3394:{\displaystyle H_{n}}
3359:
2883:
2881:{\displaystyle H_{n}}
2854:
2813:
2791:
2714:
2657:
2595:
2575:
2540:
2503:
2501:{\displaystyle y=1/x}
2473:
2449:
2387:
2374:
2342:
2069:
1687:
1658:
1558:
1534:
1532:{\displaystyle H_{n}}
1435:
1404:
1373:
1328:
1276:
1259:average case analysis
1201:
1171:
1151:
1119:
1097:
994:
926:Mathematical analysis
837:Generalized functions
522:arithmetico-geometric
363:Leibniz integral rule
117:
11823:Trigonometric series
11615:Properties of series
11462:Harmonic progression
11236:(11–12): 1062–1074.
11111:
11040:{\displaystyle \pi }
11031:
10950:Cormen et al. (2009)
10530:Mathematics Magazine
10384:. pp. 272–278.
10377:Concrete Mathematics
10189:Mathematics Magazine
10123:
9835:
9815:
9788:
9641:
9444:
9310:
9121:
9092:
8928:
8834:
8761:
8624:
8534:
8508:
8374:
8354:
8041:
8005:
7980:
7953:
7930:
7906:
7883:
7852:
7799:
7774:
7751:
7720:
7662:
7619:
7578:
7566:the solution to the
7518:
7485:
7474:holomorphic function
7446:
7307:
7281:
7277:is defined for real
7120:
7112:Leibniz formula for
6940:
6779:
6585:
6548:
6522:
6502:
6482:
6462:
6442:
6422:
6402:
6382:
6362:
6326:Analyzing algorithms
6304:
6276:
6244:
6219:
6199:
6169:
6139:
6119:
6099:
5980:
5947:
5821:
5809:, formalized as the
5793:
5776:prime number theorem
5745:
5484:
5474:prime factorizations
5450:
5279:
5263:observed that, as a
5226:
5198:
5159:
5139:
5095:
5047:
5018:
4998:
4880:
4854:
4843:{\displaystyle r=30}
4828:
4803:
4775:
4651:
4631:
4610:
4506:
4486:
4442:
4360:
4229:
4161:
4128:
4095:
4089:Bertrand's postulate
4063:
4032:
4008:
3981:
3942:
3868:
3841:
3819:
3813:of the numbers from
3793:
3771:
3742:
3712:
3677:
3648:
3591:
3561:
3485:
3406:
3378:
3348:
2865:
2843:
2802:
2670:
2608:
2584:
2549:
2520:
2478:
2458:in around 1350. The
2384:
2363:
2078:
1700:
1555:
1516:
1460:harmonic progression
1452:harmonic progression
1415:
1384:
1353:
1281:
1243:connected components
1184:
1169:{\displaystyle \ln }
1160:
1128:
1108:
991:
931:Nonstandard analysis
399:Lebesgue integration
269:Rules and identities
40:
11803:Formal power series
11065:1943RSPSA.182..113S
10790:Quality Engineering
10733:Teaching Statistics
10641:2010ScChA..53.2561T
8130:Shell-Gellasch, Amy
7878:for values between
7646:conjectured by the
7500:{\displaystyle x=1}
7461:{\displaystyle x=1}
6537:{\displaystyle k+2}
6186:{\displaystyle n/k}
6156:{\displaystyle k/n}
5781:Another problem in
5741:, and the constant
4869:{\displaystyle n=3}
4457:{\displaystyle n=3}
4192:terminating decimal
4080:{\displaystyle n/2}
2762:
2693:
2625:
1226:, according to the
602:Cauchy condensation
404:Contour integration
130:Fundamental theorem
57:
11601:Monotonic function
11520:Fibonacci sequence
11377:Weisstein, Eric W.
11117:
11037:
10745:10.1111/test.12005
10129:
9944:Bressoud, David M.
9848:
9821:
9801:
9749:
9628:
9430:
9297:
9107:
9105:
9076:"Corollary III of
9009:
9001:
8986:
8971:
8956:
8941:
8909:
8907:
8892:
8877:
8862:
8847:
8767:
8747:
8727:
8712:
8679:
8664:
8649:
8610:
8608:
8593:
8570:
8555:
8520:
8494:
8483:
8468:
8453:
8432:
8417:
8402:
8360:
8063:
8026:
8018:
7989:
7965:{\displaystyle -3}
7962:
7936:
7912:
7895:{\displaystyle -1}
7892:
7867:
7865:
7814:
7812:
7786:{\displaystyle -1}
7783:
7763:{\displaystyle +1}
7760:
7733:
7706:
7648:Riemann hypothesis
7634:
7632:
7593:
7554:
7497:
7458:
7432:
7293:
7250:
7098:
6901:
6773:
6724:
6571:
6569:
6534:
6508:
6488:
6468:
6448:
6428:
6408:
6388:
6368:
6342:
6310:
6289:
6260:
6225:
6205:
6183:
6153:
6125:
6105:
6072:
6056:Collecting coupons
6034:
5962:
5960:
5933:
5799:
5751:
5727:
5707:
5610:
5557:
5510:
5458:
5436:
5406:
5333:
5232:
5211:
5184:
5172:
5145:
5125:
5123:
5108:
5077:
5075:
5060:
5033:
5031:
5004:
4988:
4953:
4942:
4927:
4912:
4893:
4866:
4840:
4809:
4788:
4761:
4637:
4616:
4596:
4588:
4556:
4524:
4492:
4466:
4454:
4400:
4337:
4218:is defined as the
4212:
4180:
4147:
4114:
4077:
4045:
4014:
3993:
3963:
3927:
3925:
3854:
3825:
3799:
3777:
3765:in the range from
3755:
3725:
3696:
3664:goes to infinity.
3654:
3634:
3573:
3547:
3475:logarithmic growth
3456:
3391:
3354:
2878:
2849:
2808:
2786:
2748:
2673:
2652:
2611:
2590:
2570:
2568:
2535:
2533:
2510:
2498:
2444:
2369:
2337:
2335:
2064:
2062:
2046:
2004:
1980:
1956:
1914:
1690:
1653:
1529:
1487:Richard Swineshead
1430:
1428:
1399:
1397:
1368:
1366:
1334:
1323:
1315:
1300:
1196:
1166:
1146:
1114:
1092:
774:Partial derivative
703:generalized Stokes
597:Alternating series
478:Reduction formulae
467:Heaviside's method
448:tangent half-angle
435:Cylindrical shells
358:Integral transform
353:Lists of integrals
157:Mean value theorem
112:
43:
11902:
11901:
11833:Generating series
11781:
11780:
11753:1 − 2 + 4 − 8 + ⋯
11748:1 + 2 + 4 + 8 + ⋯
11743:1 − 2 + 3 − 4 + ⋯
11738:1 + 2 + 3 + 4 + ⋯
11728:1 + 1 + 1 + 1 + ⋯
11678:
11677:
11606:Periodic sequence
11575:
11574:
11560:Triangular number
11550:Pentagonal number
11530:Heptagonal number
11515:Complete sequence
11437:Integer sequences
11380:"Harmonic Series"
11120:{\displaystyle L}
10917:Rivest, Ronald L.
10909:Cormen, Thomas H.
10845:978-1-107-11850-8
10391:978-0-201-55802-9
10271:Singmaster, David
10132:{\displaystyle p}
10053:978-0-691-14133-6
9986:Wrench, J. W. Jr.
9959:978-0-88385-747-2
9824:{\displaystyle H}
9735:
9693:
9626:
9554:
9482:
9425:
9383:
9341:
9285:
9264:
9229:
9208:
9173:
9152:
9132:
9104:
9078:De seriebus varia
9072:Bernoulli, Johann
9000:
8985:
8970:
8955:
8940:
8906:
8891:
8876:
8861:
8846:
8770:{\displaystyle S}
8726:
8711:
8678:
8663:
8648:
8607:
8592:
8569:
8554:
8482:
8467:
8452:
8431:
8416:
8401:
8363:{\displaystyle S}
8273:978-1-4419-6052-8
8240:] (in Latin).
8215:978-0-226-32783-9
8202:Hersey, George L.
8143:978-0-88385-984-1
8017:
7939:{\displaystyle 3}
7915:{\displaystyle 1}
7864:
7811:
7716:where the values
7701:
7631:
7611:irrational number
7421:
7401:
7381:
7361:
7245:
7226:
7213:
7200:
7181:
7071:
7012:
6994:
6964:
6951:
6927:. Its sum is the
6893:
6880:
6867:
6854:
6835:
6652:
6568:
6511:{\displaystyle k}
6491:{\displaystyle y}
6471:{\displaystyle x}
6451:{\displaystyle y}
6431:{\displaystyle x}
6411:{\displaystyle k}
6398:are separated by
6391:{\displaystyle y}
6371:{\displaystyle x}
6322:harmonic number.
6313:{\displaystyle n}
6228:{\displaystyle n}
6208:{\displaystyle k}
6128:{\displaystyle n}
6108:{\displaystyle k}
6029:
5959:
5915:
5884:
5867:
5832:
5802:{\displaystyle n}
5772:Mertens' theorems
5754:{\displaystyle K}
5716:
5690:
5674:
5649:
5624:
5593:
5588:
5540:
5535:
5493:
5431:
5389:
5373:
5353:
5316:
5311:
5235:{\displaystyle n}
5171:
5148:{\displaystyle n}
5122:
5107:
5074:
5059:
5030:
5007:{\displaystyle n}
4941:
4926:
4911:
4892:
4812:{\displaystyle n}
4746:
4727:
4697:
4667:
4640:{\displaystyle n}
4619:{\displaystyle r}
4587:
4555:
4523:
4495:{\displaystyle n}
4424:Crossing a desert
4332:
4260:
4017:{\displaystyle M}
3924:
3828:{\displaystyle n}
3802:{\displaystyle M}
3780:{\displaystyle n}
3657:{\displaystyle n}
3532:
3451:
3357:{\displaystyle n}
3344:Adding the first
3342:
3341:
2852:{\displaystyle n}
2811:{\displaystyle n}
2771:
2743:
2702:
2634:
2602:improper integral
2593:{\displaystyle x}
2567:
2532:
2514:improper integral
2442:
2423:
2372:{\displaystyle k}
2322:
2309:
2296:
2283:
2243:
2224:
2201:
2188:
2175:
2162:
2139:
2126:
2104:
2047:
2023:
2005:
1981:
1957:
1933:
1915:
1891:
1851:
1833:
1815:
1797:
1779:
1761:
1743:
1725:
1645:
1632:
1619:
1606:
1587:
1427:
1396:
1365:
1314:
1299:
1178:natural logarithm
1117:{\displaystyle n}
1081:
1068:
1055:
1042:
1023:
969:
968:
849:
848:
811:
810:
779:Multiple integral
715:
714:
619:
618:
586:Direct comparison
557:Convergence tests
495:
494:
463:Partial fractions
330:
329:
240:Second derivative
11922:
11915:Divergent series
11892:
11891:
11818:Dirichlet series
11687:
11686:
11619:
11618:
11583:
11555:Polygonal number
11535:Hexagonal number
11508:
11442:
11441:
11418:
11411:
11404:
11395:
11394:
11390:
11389:
11362:
11361:
11323:
11317:
11316:
11286:
11280:
11279:
11245:
11225:
11219:
11218:
11216:
11215:
11209:
11203:. Archived from
11176:
11167:
11161:
11160:
11126:
11124:
11123:
11118:
11101:
11095:
11094:
11076:
11059:(989): 113–129.
11046:
11044:
11043:
11038:
11019:
11013:
11012:
10968:
10959:
10953:
10947:
10941:
10940:
10905:
10899:
10898:
10864:
10858:
10857:
10820:
10814:
10813:
10785:
10779:
10778:
10763:
10757:
10756:
10728:
10722:
10721:
10696:(251): 328–331.
10685:
10679:
10678:
10660:
10635:(9): 2561–2572.
10624:
10618:
10617:
10587:
10581:
10580:
10546:
10524:
10515:
10514:
10495:
10489:
10488:
10431:
10425:
10424:
10414:
10405:
10396:
10395:
10368:Knuth, Donald E.
10360:
10354:
10353:
10320:
10314:
10312:
10287:(475): 102–126.
10266:
10257:
10256:
10228:
10222:
10221:
10183:
10177:
10176:
10166:
10138:
10136:
10135:
10130:
10116:
10110:
10108:
10083:(537): 515–519.
10071:Osler, Thomas J.
10067:
10058:
10057:
10035:
10024:
10023:
9978:
9972:
9971:
9940:
9931:
9930:
9903:
9897:
9891:
9881:
9872:
9859:
9857:
9855:
9854:
9849:
9847:
9846:
9830:
9828:
9827:
9822:
9810:
9808:
9807:
9802:
9800:
9799:
9782:
9773:Knuth, Donald E.
9769:
9760:
9758:
9756:
9755:
9750:
9736:
9734:
9720:
9717:
9712:
9694:
9692:
9669:
9666:
9661:
9637:
9635:
9634:
9629:
9627:
9625:
9602:
9599:
9594:
9578:
9573:
9555:
9553:
9530:
9527:
9522:
9506:
9501:
9483:
9475:
9472:
9467:
9439:
9437:
9436:
9431:
9426:
9424:
9389:
9384:
9382:
9347:
9342:
9340:
9317:
9306:
9304:
9303:
9298:
9293:
9292:
9286:
9284:
9270:
9265:
9263:
9249:
9247:
9246:
9237:
9236:
9230:
9228:
9214:
9209:
9207:
9193:
9191:
9190:
9181:
9180:
9174:
9172:
9158:
9153:
9145:
9143:
9142:
9133:
9125:
9116:
9114:
9113:
9108:
9106:
9097:
9087:
9068:
9062:
9061:
9031:
9022:
9018:
9016:
9015:
9010:
9002:
8993:
8987:
8978:
8972:
8963:
8957:
8948:
8942:
8933:
8918:
8916:
8915:
8910:
8908:
8899:
8893:
8884:
8878:
8869:
8863:
8854:
8848:
8839:
8823:
8810:Bernoulli, Jacob
8806:
8800:
8799:
8788:Bernoulli, Jacob
8784:
8778:
8776:
8774:
8773:
8768:
8756:
8754:
8753:
8748:
8728:
8719:
8713:
8704:
8680:
8671:
8665:
8656:
8650:
8641:
8619:
8617:
8616:
8611:
8609:
8600:
8594:
8591:
8577:
8571:
8562:
8556:
8553:
8539:
8529:
8527:
8526:
8521:
8503:
8501:
8500:
8495:
8484:
8475:
8469:
8460:
8454:
8445:
8433:
8424:
8418:
8409:
8403:
8394:
8369:
8367:
8366:
8361:
8349:
8330:
8324:
8323:
8296:Derbyshire, John
8292:
8286:
8285:
8248:
8242:
8241:
8226:
8220:
8219:
8198:
8192:
8191:
8161:
8148:
8147:
8125:
8103:
8102:
8099:
8096:
8093:
8074:
8072:
8070:
8069:
8064:
8062:
8061:
8035:
8033:
8032:
8027:
8019:
8010:
8000:
7998:
7996:
7995:
7990:
7973:
7971:
7969:
7968:
7963:
7946:
7945:
7943:
7942:
7937:
7923:
7921:
7919:
7918:
7913:
7901:
7899:
7898:
7893:
7877:
7876:
7874:
7873:
7868:
7866:
7857:
7825:
7823:
7821:
7820:
7815:
7813:
7804:
7792:
7790:
7789:
7784:
7769:
7767:
7766:
7761:
7742:
7740:
7739:
7734:
7732:
7731:
7715:
7713:
7712:
7707:
7702:
7697:
7696:
7687:
7684:
7679:
7645:
7643:
7641:
7640:
7635:
7633:
7624:
7604:
7602:
7600:
7599:
7594:
7572:Apéry's constant
7565:
7563:
7561:
7560:
7555:
7550:
7545:
7544:
7508:
7506:
7504:
7503:
7498:
7467:
7465:
7464:
7459:
7441:
7439:
7438:
7433:
7422:
7420:
7419:
7407:
7402:
7400:
7399:
7387:
7382:
7380:
7379:
7367:
7362:
7360:
7359:
7347:
7344:
7339:
7302:
7300:
7299:
7294:
7259:
7257:
7256:
7251:
7246:
7238:
7227:
7219:
7214:
7206:
7201:
7193:
7182:
7180:
7166:
7165:
7164:
7145:
7142:
7137:
7115:
7107:
7105:
7104:
7099:
7094:
7093:
7072:
7070:
7059:
7042:
7041:
7029:
7028:
7013:
7011:
7000:
6995:
6993:
6976:
6965:
6957:
6952:
6944:
6911:is known as the
6910:
6908:
6907:
6902:
6894:
6886:
6881:
6873:
6868:
6860:
6855:
6847:
6836:
6831:
6830:
6829:
6804:
6801:
6796:
6733:
6731:
6730:
6725:
6696:
6695:
6682:
6671:
6653:
6651:
6637:
6634:
6623:
6607:
6602:
6580:
6578:
6577:
6572:
6570:
6567:
6553:
6543:
6541:
6540:
6535:
6517:
6515:
6514:
6509:
6497:
6495:
6494:
6489:
6477:
6475:
6474:
6469:
6457:
6455:
6454:
6449:
6437:
6435:
6434:
6429:
6417:
6415:
6414:
6409:
6397:
6395:
6394:
6389:
6377:
6375:
6374:
6369:
6321:
6319:
6317:
6316:
6311:
6298:
6296:
6295:
6290:
6288:
6287:
6271:
6269:
6267:
6266:
6261:
6259:
6258:
6237:
6234:
6232:
6231:
6226:
6214:
6212:
6211:
6206:
6194:
6192:
6190:
6189:
6184:
6179:
6162:
6160:
6159:
6154:
6149:
6134:
6132:
6131:
6126:
6114:
6112:
6111:
6106:
6043:
6041:
6040:
6035:
6030:
6025:
6023:
5971:
5969:
5968:
5963:
5961:
5952:
5942:
5940:
5939:
5934:
5929:
5928:
5916:
5908:
5905:
5900:
5885:
5877:
5872:
5868:
5860:
5853:
5848:
5833:
5825:
5815:divisor function
5808:
5806:
5805:
5800:
5768:double logarithm
5760:
5758:
5757:
5752:
5736:
5734:
5733:
5728:
5717:
5709:
5706:
5705:
5686:
5682:
5675:
5673:
5672:
5671:
5655:
5650:
5648:
5647:
5646:
5630:
5625:
5617:
5609:
5608:
5589:
5587:
5583:
5565:
5556:
5555:
5536:
5534:
5530:
5512:
5509:
5508:
5478:geometric series
5470:distributive law
5467:
5465:
5464:
5459:
5457:
5445:
5443:
5442:
5437:
5432:
5430:
5426:
5408:
5405:
5404:
5385:
5381:
5374:
5372:
5371:
5359:
5354:
5346:
5332:
5331:
5312:
5304:
5301:
5296:
5243:
5241:
5239:
5238:
5233:
5220:
5218:
5217:
5212:
5210:
5209:
5193:
5191:
5190:
5185:
5183:
5182:
5173:
5164:
5154:
5152:
5151:
5146:
5134:
5132:
5131:
5126:
5124:
5115:
5109:
5100:
5086:
5084:
5083:
5078:
5076:
5067:
5061:
5052:
5042:
5040:
5039:
5034:
5032:
5023:
5013:
5011:
5010:
5005:
4962:
4960:
4959:
4954:
4943:
4934:
4928:
4919:
4913:
4904:
4901:
4900:
4894:
4885:
4875:
4873:
4872:
4867:
4849:
4847:
4846:
4841:
4820:
4818:
4816:
4815:
4810:
4797:
4795:
4794:
4789:
4787:
4786:
4770:
4768:
4767:
4762:
4757:
4756:
4747:
4739:
4728:
4726:
4703:
4698:
4696:
4673:
4668:
4666:
4655:
4646:
4644:
4643:
4638:
4625:
4623:
4622:
4617:
4605:
4603:
4602:
4597:
4589:
4586:
4563:
4557:
4554:
4531:
4525:
4522:
4511:
4501:
4499:
4498:
4493:
4463:
4461:
4460:
4455:
4411:
4409:
4407:
4406:
4401:
4393:
4392:
4346:
4344:
4343:
4338:
4333:
4331:
4317:
4307:
4298:
4293:
4292:
4274:
4273:
4261:
4259:
4248:
4216:digamma function
4208:digamma function
4189:
4187:
4186:
4181:
4173:
4172:
4156:
4154:
4153:
4148:
4140:
4139:
4123:
4121:
4120:
4115:
4107:
4106:
4086:
4084:
4083:
4078:
4073:
4054:
4052:
4051:
4046:
4044:
4043:
4023:
4021:
4020:
4015:
4004:
4002:
4000:
3999:
3994:
3974:
3972:
3970:
3969:
3964:
3962:
3961:
3952:
3936:
3934:
3933:
3928:
3926:
3920:
3916:
3907:
3903:
3898:
3880:
3879:
3863:
3861:
3860:
3855:
3853:
3852:
3836:
3834:
3832:
3831:
3826:
3808:
3806:
3805:
3800:
3788:
3786:
3784:
3783:
3778:
3764:
3762:
3761:
3756:
3754:
3753:
3734:
3732:
3731:
3726:
3724:
3723:
3707:
3705:
3703:
3702:
3697:
3689:
3688:
3663:
3661:
3660:
3655:
3643:
3641:
3640:
3635:
3633:
3632:
3620:
3609:
3608:
3582:
3580:
3579:
3574:
3556:
3554:
3553:
3548:
3546:
3545:
3533:
3531:
3520:
3497:
3496:
3465:
3463:
3462:
3457:
3452:
3444:
3441:
3436:
3418:
3417:
3402:
3400:
3398:
3397:
3392:
3390:
3389:
3363:
3361:
3360:
3355:
3335:
3313:
3291:
3269:
3247:
3225:
3203:
3181:
3159:
3137:
3115:
3093:
3071:
3049:
3027:
3005:
2983:
2961:
2939:
2917:
2911:
2887:
2885:
2884:
2879:
2877:
2876:
2858:
2856:
2855:
2850:
2837:
2836:
2818:
2817:
2815:
2814:
2809:
2795:
2793:
2792:
2787:
2772:
2764:
2761:
2756:
2744:
2736:
2733:
2728:
2703:
2695:
2692:
2681:
2661:
2659:
2658:
2653:
2635:
2627:
2624:
2619:
2599:
2597:
2596:
2591:
2579:
2577:
2576:
2571:
2569:
2560:
2544:
2542:
2541:
2536:
2534:
2525:
2507:
2505:
2504:
2499:
2494:
2453:
2451:
2450:
2445:
2443:
2435:
2424:
2416:
2413:
2412:
2411:
2401:
2380:
2378:
2376:
2375:
2370:
2346:
2344:
2343:
2338:
2336:
2323:
2315:
2310:
2302:
2297:
2289:
2284:
2276:
2266:
2261:
2249:
2245:
2244:
2236:
2225:
2217:
2207:
2203:
2202:
2194:
2189:
2181:
2176:
2168:
2163:
2155:
2145:
2141:
2140:
2132:
2127:
2119:
2109:
2105:
2097:
2084:
2073:
2071:
2070:
2065:
2063:
2050:
2048:
2045:
2044:
2034:
2026:
2024:
2016:
2008:
2006:
2003:
2002:
1992:
1984:
1982:
1979:
1978:
1968:
1960:
1958:
1955:
1954:
1944:
1936:
1934:
1926:
1918:
1916:
1913:
1912:
1902:
1894:
1892:
1884:
1869:
1854:
1852:
1844:
1836:
1834:
1826:
1818:
1816:
1808:
1800:
1798:
1790:
1782:
1780:
1772:
1764:
1762:
1754:
1746:
1744:
1736:
1728:
1726:
1718:
1669:divergent series
1662:
1660:
1659:
1654:
1646:
1638:
1633:
1625:
1620:
1612:
1607:
1599:
1588:
1580:
1577:
1572:
1538:
1536:
1535:
1530:
1528:
1527:
1510:harmonic numbers
1503:Johann Bernoulli
1491:geometric series
1441:
1439:
1437:
1436:
1431:
1429:
1420:
1410:
1408:
1406:
1405:
1400:
1398:
1389:
1379:
1377:
1375:
1374:
1369:
1367:
1358:
1332:
1330:
1329:
1324:
1316:
1307:
1301:
1292:
1212:divergent series
1205:
1203:
1202:
1197:
1175:
1173:
1172:
1167:
1155:
1153:
1152:
1147:
1123:
1121:
1120:
1115:
1101:
1099:
1098:
1093:
1082:
1074:
1069:
1061:
1056:
1048:
1043:
1035:
1024:
1016:
1013:
1008:
961:
954:
947:
895:
860:
826:
825:
822:
789:Surface integral
732:
731:
728:
636:
635:
632:
592:Limit comparison
512:
511:
508:
394:Riemann integral
347:
346:
343:
303:L'Hôpital's rule
260:Taylor's theorem
181:
180:
177:
121:
119:
118:
113:
65:
56:
51:
21:
20:
11930:
11929:
11925:
11924:
11923:
11921:
11920:
11919:
11905:
11904:
11903:
11898:
11880:
11837:
11786:Kinds of series
11777:
11716:
11683:Explicit series
11674:
11648:
11610:
11596:Cauchy sequence
11584:
11571:
11525:Figurate number
11502:
11496:
11487:Powers of three
11431:
11422:
11371:
11366:
11365:
11324:
11320:
11287:
11283:
11226:
11222:
11213:
11211:
11207:
11193:10.2307/3647827
11174:
11168:
11164:
11112:
11109:
11108:
11102:
11098:
11032:
11029:
11028:
11020:
11016:
10966:
10960:
10956:
10948:
10944:
10937:
10921:Stein, Clifford
10906:
10902:
10887:
10865:
10861:
10846:
10821:
10817:
10786:
10782:
10764:
10760:
10729:
10725:
10702:10.2307/3607889
10686:
10682:
10625:
10621:
10588:
10584:
10525:
10518:
10499:Euler, Leonhard
10496:
10492:
10432:
10428:
10412:
10406:
10399:
10392:
10380:(2e ed.).
10372:Patashnik, Oren
10361:
10357:
10321:
10317:
10293:10.2307/3620384
10267:
10260:
10229:
10225:
10184:
10180:
10124:
10121:
10120:
10117:
10113:
10068:
10061:
10054:
10036:
10027:
9982:Boas, R. P. Jr.
9979:
9975:
9960:
9941:
9934:
9904:
9900:
9879:
9873:
9862:
9842:
9838:
9836:
9833:
9832:
9816:
9813:
9812:
9811:... The letter
9795:
9791:
9789:
9786:
9785:
9770:
9763:
9724:
9719:
9713:
9702:
9673:
9668:
9662:
9651:
9642:
9639:
9638:
9606:
9601:
9595:
9584:
9574:
9563:
9534:
9529:
9523:
9512:
9502:
9491:
9474:
9468:
9457:
9445:
9442:
9441:
9393:
9388:
9351:
9346:
9321:
9316:
9311:
9308:
9307:
9288:
9287:
9274:
9269:
9253:
9248:
9242:
9241:
9232:
9231:
9218:
9213:
9197:
9192:
9186:
9185:
9176:
9175:
9162:
9157:
9144:
9138:
9137:
9124:
9122:
9119:
9118:
9095:
9093:
9090:
9089:
9069:
9065:
9032:
9025:
8991:
8976:
8961:
8946:
8931:
8929:
8926:
8925:
8897:
8882:
8867:
8852:
8837:
8835:
8832:
8831:
8824:
8807:
8803:
8785:
8781:
8762:
8759:
8758:
8717:
8702:
8669:
8654:
8639:
8625:
8622:
8621:
8598:
8581:
8575:
8560:
8543:
8537:
8535:
8532:
8531:
8509:
8506:
8505:
8473:
8458:
8443:
8422:
8407:
8392:
8375:
8372:
8371:
8355:
8352:
8351:
8334:Mengoli, Pietro
8331:
8327:
8312:
8293:
8289:
8274:
8252:Stillwell, John
8249:
8245:
8227:
8223:
8216:
8199:
8195:
8180:10.2307/2687471
8162:
8151:
8144:
8126:
8119:
8114:
8100:
8097:
8094:
8091:
8089:
8086:
8080:
8054:
8050:
8042:
8039:
8038:
8037:
8008:
8006:
8003:
8002:
7981:
7978:
7977:
7975:
7954:
7951:
7950:
7948:
7931:
7928:
7927:
7925:
7907:
7904:
7903:
7884:
7881:
7880:
7879:
7855:
7853:
7850:
7849:
7847:
7840:random variable
7802:
7800:
7797:
7796:
7794:
7775:
7772:
7771:
7752:
7749:
7748:
7727:
7723:
7721:
7718:
7717:
7692:
7688:
7686:
7680:
7669:
7663:
7660:
7659:
7656:
7622:
7620:
7617:
7616:
7614:
7579:
7576:
7575:
7574:
7546:
7540:
7536:
7519:
7516:
7515:
7514:
7486:
7483:
7482:
7480:
7478:complex numbers
7447:
7444:
7443:
7415:
7411:
7406:
7395:
7391:
7386:
7375:
7371:
7366:
7355:
7351:
7346:
7340:
7329:
7308:
7305:
7304:
7282:
7279:
7278:
7271:
7265:
7237:
7218:
7205:
7192:
7167:
7160:
7156:
7146:
7144:
7138:
7127:
7121:
7118:
7117:
7113:
7086:
7082:
7063:
7058:
7037:
7033:
7021:
7017:
7004:
6999:
6980:
6975:
6956:
6943:
6941:
6938:
6937:
6885:
6872:
6859:
6846:
6819:
6815:
6805:
6803:
6797:
6786:
6780:
6777:
6776:
6765:
6759:
6754:
6748:
6691:
6687:
6672:
6661:
6641:
6636:
6624:
6613:
6603:
6592:
6586:
6583:
6582:
6557:
6551:
6549:
6546:
6545:
6523:
6520:
6519:
6503:
6500:
6499:
6483:
6480:
6479:
6463:
6460:
6459:
6443:
6440:
6439:
6423:
6420:
6419:
6403:
6400:
6399:
6383:
6380:
6379:
6363:
6360:
6359:
6334:
6328:
6305:
6302:
6301:
6300:
6283:
6279:
6277:
6274:
6273:
6254:
6250:
6245:
6242:
6241:
6239:
6235:
6220:
6217:
6216:
6200:
6197:
6196:
6175:
6170:
6167:
6166:
6164:
6145:
6140:
6137:
6136:
6120:
6117:
6116:
6100:
6097:
6096:
6085:quality control
6064:
6058:
6024:
6019:
5981:
5978:
5977:
5950:
5948:
5945:
5944:
5924:
5920:
5907:
5901:
5890:
5876:
5859:
5855:
5849:
5838:
5824:
5822:
5819:
5818:
5794:
5791:
5790:
5746:
5743:
5742:
5708:
5701:
5694:
5667:
5663:
5659:
5654:
5642:
5638:
5634:
5629:
5616:
5615:
5611:
5604:
5597:
5579:
5569:
5564:
5551:
5544:
5526:
5516:
5511:
5504:
5497:
5485:
5482:
5481:
5453:
5451:
5448:
5447:
5422:
5412:
5407:
5400:
5393:
5367:
5363:
5358:
5345:
5338:
5334:
5327:
5320:
5303:
5297:
5286:
5280:
5277:
5276:
5257:
5251:
5227:
5224:
5223:
5222:
5205:
5201:
5199:
5196:
5195:
5178:
5174:
5162:
5160:
5157:
5156:
5140:
5137:
5136:
5113:
5098:
5096:
5093:
5092:
5065:
5050:
5048:
5045:
5044:
5021:
5019:
5016:
5015:
4999:
4996:
4995:
4976:
4970:
4968:Stacking blocks
4932:
4917:
4902:
4896:
4895:
4883:
4881:
4878:
4877:
4855:
4852:
4851:
4829:
4826:
4825:
4804:
4801:
4800:
4799:
4782:
4778:
4776:
4773:
4772:
4752:
4748:
4738:
4707:
4702:
4677:
4672:
4659:
4654:
4652:
4649:
4648:
4632:
4629:
4628:
4611:
4608:
4607:
4567:
4561:
4535:
4529:
4515:
4509:
4507:
4504:
4503:
4487:
4484:
4483:
4443:
4440:
4439:
4432:
4426:
4418:
4382:
4378:
4361:
4358:
4357:
4356:
4318:
4300:
4299:
4297:
4288:
4287:
4269:
4268:
4252:
4247:
4230:
4227:
4226:
4200:
4168:
4164:
4162:
4159:
4158:
4135:
4131:
4129:
4126:
4125:
4102:
4098:
4096:
4093:
4092:
4069:
4064:
4061:
4060:
4039:
4035:
4033:
4030:
4029:
4009:
4006:
4005:
3982:
3979:
3978:
3976:
3957:
3953:
3948:
3943:
3940:
3939:
3938:
3912:
3908:
3905:
3899:
3888:
3875:
3871:
3869:
3866:
3865:
3848:
3844:
3842:
3839:
3838:
3820:
3817:
3816:
3814:
3794:
3791:
3790:
3772:
3769:
3768:
3766:
3749:
3745:
3743:
3740:
3739:
3719:
3715:
3713:
3710:
3709:
3684:
3680:
3678:
3675:
3674:
3673:
3670:
3649:
3646:
3645:
3628:
3624:
3616:
3604:
3600:
3592:
3589:
3588:
3562:
3559:
3558:
3541:
3537:
3524:
3519:
3492:
3488:
3486:
3483:
3482:
3471:
3443:
3437:
3426:
3413:
3409:
3407:
3404:
3403:
3385:
3381:
3379:
3376:
3375:
3373:
3370:harmonic number
3349:
3346:
3345:
3338:
3333:
3316:
3311:
3294:
3289:
3272:
3267:
3250:
3245:
3228:
3223:
3206:
3201:
3184:
3179:
3162:
3157:
3140:
3135:
3118:
3113:
3096:
3091:
3074:
3069:
3052:
3047:
3030:
3025:
3008:
3003:
2986:
2981:
2964:
2959:
2942:
2937:
2920:
2915:
2909:
2872:
2868:
2866:
2863:
2862:
2844:
2841:
2840:
2835:
2833:Harmonic number
2829:
2803:
2800:
2799:
2797:
2763:
2757:
2752:
2735:
2729:
2718:
2694:
2682:
2677:
2671:
2668:
2667:
2626:
2620:
2615:
2609:
2606:
2605:
2585:
2582:
2581:
2558:
2550:
2547:
2546:
2523:
2521:
2518:
2517:
2490:
2479:
2476:
2475:
2468:
2434:
2415:
2407:
2403:
2402:
2391:
2385:
2382:
2381:
2364:
2361:
2360:
2356:
2350:comparison test
2334:
2333:
2314:
2301:
2288:
2275:
2267:
2265:
2260:
2257:
2256:
2235:
2216:
2215:
2211:
2193:
2180:
2167:
2154:
2153:
2149:
2131:
2118:
2117:
2113:
2096:
2092:
2081:
2079:
2076:
2075:
2061:
2060:
2049:
2040:
2039:
2033:
2025:
2015:
2007:
1998:
1997:
1991:
1983:
1974:
1973:
1967:
1959:
1950:
1949:
1943:
1935:
1925:
1917:
1908:
1907:
1901:
1893:
1883:
1876:
1868:
1865:
1864:
1853:
1843:
1835:
1825:
1817:
1807:
1799:
1789:
1781:
1771:
1763:
1753:
1745:
1735:
1727:
1717:
1710:
1703:
1701:
1698:
1697:
1682:
1680:Comparison test
1637:
1624:
1611:
1598:
1579:
1573:
1562:
1556:
1553:
1552:
1549:
1523:
1519:
1517:
1514:
1513:
1499:Jacob Bernoulli
1418:
1416:
1413:
1412:
1411:
1387:
1385:
1382:
1381:
1380:
1356:
1354:
1351:
1350:
1349:
1305:
1290:
1282:
1279:
1278:
1271:
1185:
1182:
1181:
1161:
1158:
1157:
1129:
1126:
1125:
1109:
1106:
1105:
1073:
1060:
1047:
1034:
1015:
1009:
998:
992:
989:
988:
981:infinite series
977:harmonic series
965:
936:
935:
921:Integration Bee
896:
893:
886:
885:
861:
858:
851:
850:
823:
820:
813:
812:
794:Volume integral
729:
724:
717:
716:
633:
628:
621:
620:
590:
509:
504:
497:
496:
488:Risch algorithm
458:Euler's formula
344:
339:
332:
331:
313:General Leibniz
196:generalizations
178:
173:
166:
152:Rolle's theorem
147:
122:
58:
52:
47:
41:
38:
37:
19:
12:
11:
5:
11928:
11918:
11917:
11900:
11899:
11897:
11896:
11885:
11882:
11881:
11879:
11878:
11873:
11868:
11863:
11858:
11853:
11847:
11845:
11839:
11838:
11836:
11835:
11830:
11828:Fourier series
11825:
11820:
11815:
11813:Puiseux series
11810:
11808:Laurent series
11805:
11800:
11795:
11789:
11787:
11783:
11782:
11779:
11778:
11776:
11775:
11770:
11765:
11760:
11755:
11750:
11745:
11740:
11735:
11730:
11724:
11722:
11718:
11717:
11715:
11714:
11709:
11704:
11699:
11693:
11691:
11684:
11680:
11679:
11676:
11675:
11673:
11672:
11667:
11662:
11656:
11654:
11650:
11649:
11647:
11646:
11641:
11636:
11631:
11625:
11623:
11616:
11612:
11611:
11609:
11608:
11603:
11598:
11592:
11590:
11586:
11585:
11578:
11576:
11573:
11572:
11570:
11569:
11568:
11567:
11557:
11552:
11547:
11542:
11537:
11532:
11527:
11522:
11517:
11511:
11509:
11498:
11497:
11495:
11494:
11489:
11484:
11479:
11474:
11469:
11464:
11459:
11454:
11448:
11446:
11439:
11433:
11432:
11421:
11420:
11413:
11406:
11398:
11392:
11391:
11370:
11369:External links
11367:
11364:
11363:
11336:(6): 525–540.
11318:
11299:(5): 372–374.
11281:
11220:
11187:(5): 407–416.
11162:
11116:
11096:
11036:
11014:
10979:(5): 442–448.
10954:
10942:
10935:
10900:
10885:
10859:
10844:
10815:
10796:(2): 168–181.
10780:
10775:Stand-up maths
10758:
10723:
10680:
10619:
10606:10.4171/EM/268
10582:
10537:(5): 355–359.
10516:
10490:
10463:(9): 763–787.
10447:Winkler, Peter
10443:Thorup, Mikkel
10435:Paterson, Mike
10426:
10423:(10): 411–412.
10397:
10390:
10382:Addison-Wesley
10364:Graham, Ronald
10355:
10336:(5): 493–501.
10315:
10269:Hadley, John;
10258:
10223:
10196:(3): 176–179.
10178:
10128:
10111:
10059:
10052:
10025:
9998:(8): 864–870.
9973:
9958:
9932:
9918:(4): 717–719.
9898:
9860:
9845:
9841:
9820:
9798:
9794:
9761:
9748:
9745:
9742:
9739:
9733:
9730:
9727:
9723:
9716:
9711:
9708:
9705:
9701:
9697:
9691:
9688:
9685:
9682:
9679:
9676:
9672:
9665:
9660:
9657:
9654:
9650:
9646:
9624:
9621:
9618:
9615:
9612:
9609:
9605:
9598:
9593:
9590:
9587:
9583:
9577:
9572:
9569:
9566:
9562:
9558:
9552:
9549:
9546:
9543:
9540:
9537:
9533:
9526:
9521:
9518:
9515:
9511:
9505:
9500:
9497:
9494:
9490:
9486:
9481:
9478:
9471:
9466:
9463:
9460:
9456:
9452:
9449:
9429:
9423:
9420:
9417:
9414:
9411:
9408:
9405:
9402:
9399:
9396:
9392:
9387:
9381:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9350:
9345:
9339:
9336:
9333:
9330:
9327:
9324:
9320:
9315:
9296:
9291:
9283:
9280:
9277:
9273:
9268:
9262:
9259:
9256:
9252:
9245:
9240:
9235:
9227:
9224:
9221:
9217:
9212:
9206:
9203:
9200:
9196:
9189:
9184:
9179:
9171:
9168:
9165:
9161:
9156:
9151:
9148:
9141:
9136:
9131:
9128:
9103:
9100:
9063:
9023:
9021:
9020:
9008:
9005:
8999:
8996:
8990:
8984:
8981:
8975:
8969:
8966:
8960:
8954:
8951:
8945:
8939:
8936:
8922:
8905:
8902:
8896:
8890:
8887:
8881:
8875:
8872:
8866:
8860:
8857:
8851:
8845:
8842:
8801:
8779:
8766:
8746:
8743:
8740:
8737:
8734:
8731:
8725:
8722:
8716:
8710:
8707:
8701:
8698:
8695:
8692:
8689:
8686:
8683:
8677:
8674:
8668:
8662:
8659:
8653:
8647:
8644:
8638:
8635:
8632:
8629:
8606:
8603:
8597:
8590:
8587:
8584:
8580:
8574:
8568:
8565:
8559:
8552:
8549:
8546:
8542:
8519:
8516:
8513:
8493:
8490:
8487:
8481:
8478:
8472:
8466:
8463:
8457:
8451:
8448:
8442:
8439:
8436:
8430:
8427:
8421:
8415:
8412:
8406:
8400:
8397:
8391:
8388:
8385:
8382:
8379:
8359:
8325:
8310:
8287:
8272:
8243:
8230:Oresme, Nicole
8221:
8214:
8193:
8174:(3): 201–203.
8149:
8142:
8116:
8115:
8113:
8110:
8084:Kempner series
8082:Main article:
8079:
8076:
8060:
8057:
8053:
8049:
8046:
8025:
8022:
8016:
8013:
7988:
7985:
7961:
7958:
7935:
7911:
7891:
7888:
7863:
7860:
7810:
7807:
7782:
7779:
7759:
7756:
7730:
7726:
7705:
7700:
7695:
7691:
7683:
7678:
7675:
7672:
7668:
7655:
7652:
7630:
7627:
7592:
7589:
7586:
7583:
7553:
7549:
7543:
7539:
7535:
7532:
7529:
7526:
7523:
7496:
7493:
7490:
7457:
7454:
7451:
7431:
7428:
7425:
7418:
7414:
7410:
7405:
7398:
7394:
7390:
7385:
7378:
7374:
7370:
7365:
7358:
7354:
7350:
7343:
7338:
7335:
7332:
7328:
7324:
7321:
7318:
7315:
7312:
7292:
7289:
7286:
7267:Main article:
7264:
7261:
7249:
7244:
7241:
7236:
7233:
7230:
7225:
7222:
7217:
7212:
7209:
7204:
7199:
7196:
7191:
7188:
7185:
7179:
7176:
7173:
7170:
7163:
7159:
7155:
7152:
7149:
7141:
7136:
7133:
7130:
7126:
7097:
7092:
7089:
7085:
7081:
7078:
7075:
7069:
7066:
7062:
7057:
7054:
7051:
7048:
7045:
7040:
7036:
7032:
7027:
7024:
7020:
7016:
7010:
7007:
7003:
6998:
6992:
6989:
6986:
6983:
6979:
6974:
6971:
6968:
6963:
6960:
6955:
6950:
6947:
6900:
6897:
6892:
6889:
6884:
6879:
6876:
6871:
6866:
6863:
6858:
6853:
6850:
6845:
6842:
6839:
6834:
6828:
6825:
6822:
6818:
6814:
6811:
6808:
6800:
6795:
6792:
6789:
6785:
6758:
6755:
6747:
6746:Related series
6744:
6723:
6720:
6717:
6714:
6711:
6708:
6705:
6702:
6699:
6694:
6690:
6686:
6681:
6678:
6675:
6670:
6667:
6664:
6660:
6656:
6650:
6647:
6644:
6640:
6633:
6630:
6627:
6622:
6619:
6616:
6612:
6606:
6601:
6598:
6595:
6591:
6566:
6563:
6560:
6556:
6533:
6530:
6527:
6507:
6487:
6467:
6447:
6427:
6407:
6387:
6367:
6330:Main article:
6327:
6324:
6309:
6286:
6282:
6257:
6253:
6249:
6224:
6204:
6182:
6178:
6174:
6152:
6148:
6144:
6124:
6104:
6060:Main article:
6057:
6054:
6046:big O notation
6044:(expressed in
6033:
6028:
6022:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5988:
5985:
5958:
5955:
5932:
5927:
5923:
5919:
5914:
5911:
5904:
5899:
5896:
5893:
5889:
5883:
5880:
5875:
5871:
5866:
5863:
5858:
5852:
5847:
5844:
5841:
5837:
5831:
5828:
5798:
5750:
5726:
5723:
5720:
5715:
5712:
5704:
5700:
5697:
5693:
5689:
5685:
5681:
5678:
5670:
5666:
5662:
5658:
5653:
5645:
5641:
5637:
5633:
5628:
5623:
5620:
5614:
5607:
5603:
5600:
5596:
5592:
5586:
5582:
5578:
5575:
5572:
5568:
5563:
5560:
5554:
5550:
5547:
5543:
5539:
5533:
5529:
5525:
5522:
5519:
5515:
5507:
5503:
5500:
5496:
5492:
5489:
5456:
5435:
5429:
5425:
5421:
5418:
5415:
5411:
5403:
5399:
5396:
5392:
5388:
5384:
5380:
5377:
5370:
5366:
5362:
5357:
5352:
5349:
5344:
5341:
5337:
5330:
5326:
5323:
5319:
5315:
5310:
5307:
5300:
5295:
5292:
5289:
5285:
5261:Leonhard Euler
5253:Main article:
5250:
5247:
5231:
5208:
5204:
5181:
5177:
5170:
5167:
5144:
5121:
5118:
5112:
5106:
5103:
5089:center of mass
5073:
5070:
5064:
5058:
5055:
5029:
5026:
5003:
4972:Main article:
4969:
4966:
4952:
4949:
4946:
4940:
4937:
4931:
4925:
4922:
4916:
4910:
4907:
4899:
4891:
4888:
4865:
4862:
4859:
4839:
4836:
4833:
4808:
4785:
4781:
4760:
4755:
4751:
4745:
4742:
4737:
4734:
4731:
4725:
4722:
4719:
4716:
4713:
4710:
4706:
4701:
4695:
4692:
4689:
4686:
4683:
4680:
4676:
4671:
4665:
4662:
4658:
4636:
4615:
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4582:
4579:
4576:
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4570:
4566:
4560:
4553:
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4544:
4541:
4538:
4534:
4528:
4521:
4518:
4514:
4491:
4453:
4450:
4447:
4428:Main article:
4425:
4422:
4417:
4414:
4399:
4396:
4391:
4388:
4385:
4381:
4377:
4374:
4371:
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4365:
4336:
4330:
4327:
4324:
4321:
4316:
4313:
4310:
4306:
4303:
4296:
4291:
4286:
4283:
4280:
4277:
4272:
4267:
4264:
4258:
4255:
4251:
4246:
4243:
4240:
4237:
4234:
4224:gamma function
4199:
4196:
4179:
4176:
4171:
4167:
4146:
4143:
4138:
4134:
4113:
4110:
4105:
4101:
4076:
4072:
4068:
4042:
4038:
4013:
3992:
3989:
3986:
3960:
3956:
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3947:
3923:
3919:
3915:
3911:
3902:
3897:
3894:
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3722:
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3500:
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3432:
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3412:
3388:
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3336:
3331:
3328:
3325:
3322:
3318:
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3309:
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3300:
3296:
3295:
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3287:
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3278:
3274:
3273:
3270:
3265:
3262:
3259:
3256:
3252:
3251:
3248:
3243:
3240:
3237:
3234:
3230:
3229:
3226:
3221:
3218:
3215:
3212:
3208:
3207:
3204:
3199:
3196:
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3190:
3186:
3185:
3182:
3177:
3174:
3171:
3168:
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3141:
3138:
3133:
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3124:
3120:
3119:
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3111:
3108:
3105:
3102:
3098:
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3064:
3061:
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3039:
3036:
3032:
3031:
3028:
3023:
3020:
3017:
3014:
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3009:
3006:
3001:
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2979:
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2932:
2929:
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2922:
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2918:
2913:
2907:
2904:
2900:
2899:
2898:relative size
2896:
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2888:
2875:
2871:
2859:
2848:
2831:Main article:
2828:
2825:
2807:
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2776:
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2760:
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2676:
2651:
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2639:
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2589:
2566:
2563:
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2531:
2528:
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2239:
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2210:
2206:
2200:
2197:
2192:
2187:
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2171:
2166:
2161:
2158:
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2144:
2138:
2135:
2130:
2125:
2122:
2116:
2112:
2108:
2103:
2100:
2095:
2091:
2088:
2085:
2083:
2059:
2056:
2053:
2051:
2043:
2037:
2032:
2029:
2027:
2022:
2019:
2014:
2011:
2009:
2001:
1995:
1990:
1987:
1985:
1977:
1971:
1966:
1963:
1961:
1953:
1947:
1942:
1939:
1937:
1932:
1929:
1924:
1921:
1919:
1911:
1905:
1900:
1897:
1895:
1890:
1887:
1882:
1879:
1877:
1875:
1872:
1867:
1866:
1863:
1860:
1857:
1855:
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1832:
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1814:
1811:
1806:
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1801:
1796:
1793:
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1785:
1783:
1778:
1775:
1770:
1767:
1765:
1760:
1757:
1752:
1749:
1747:
1742:
1739:
1734:
1731:
1729:
1724:
1721:
1716:
1713:
1711:
1709:
1706:
1705:
1681:
1678:
1665:unit fractions
1652:
1649:
1644:
1641:
1636:
1631:
1628:
1623:
1618:
1615:
1610:
1605:
1602:
1597:
1594:
1591:
1586:
1583:
1576:
1571:
1568:
1565:
1561:
1548:
1545:
1526:
1522:
1495:Pietro Mengoli
1454:; the phrases
1426:
1423:
1395:
1392:
1364:
1361:
1322:
1319:
1313:
1310:
1304:
1298:
1295:
1289:
1286:
1270:
1267:
1195:
1192:
1189:
1165:
1145:
1142:
1139:
1136:
1133:
1113:
1091:
1088:
1085:
1080:
1077:
1072:
1067:
1064:
1059:
1054:
1051:
1046:
1041:
1038:
1033:
1030:
1027:
1022:
1019:
1012:
1007:
1004:
1001:
997:
985:unit fractions
967:
966:
964:
963:
956:
949:
941:
938:
937:
934:
933:
928:
923:
918:
916:List of topics
913:
908:
903:
897:
892:
891:
888:
887:
884:
883:
878:
873:
868:
862:
857:
856:
853:
852:
847:
846:
845:
844:
839:
834:
824:
819:
818:
815:
814:
809:
808:
807:
806:
801:
796:
791:
786:
781:
776:
768:
767:
763:
762:
761:
760:
755:
750:
745:
737:
736:
730:
723:
722:
719:
718:
713:
712:
711:
710:
705:
700:
695:
690:
685:
677:
676:
672:
671:
670:
669:
664:
659:
654:
649:
644:
634:
627:
626:
623:
622:
617:
616:
615:
614:
609:
604:
599:
594:
588:
583:
578:
573:
568:
560:
559:
553:
552:
551:
550:
545:
540:
535:
530:
525:
510:
503:
502:
499:
498:
493:
492:
491:
490:
485:
480:
475:
473:Changing order
470:
460:
455:
437:
432:
427:
419:
418:
417:Integration by
414:
413:
412:
411:
406:
401:
396:
391:
381:
379:Antiderivative
373:
372:
368:
367:
366:
365:
360:
355:
345:
338:
337:
334:
333:
328:
327:
326:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
272:
271:
265:
264:
263:
262:
257:
252:
247:
242:
237:
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220:
219:
214:
209:
199:
186:
185:
179:
172:
171:
168:
167:
165:
164:
159:
154:
148:
146:
145:
140:
134:
133:
132:
124:
123:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
74:
71:
68:
64:
61:
55:
50:
46:
36:
33:
32:
26:
25:
17:
9:
6:
4:
3:
2:
11927:
11916:
11913:
11912:
11910:
11895:
11887:
11886:
11883:
11877:
11874:
11872:
11869:
11867:
11864:
11862:
11859:
11857:
11854:
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11829:
11826:
11824:
11821:
11819:
11816:
11814:
11811:
11809:
11806:
11804:
11801:
11799:
11796:
11794:
11793:Taylor series
11791:
11790:
11788:
11784:
11774:
11771:
11769:
11766:
11764:
11761:
11759:
11756:
11754:
11751:
11749:
11746:
11744:
11741:
11739:
11736:
11734:
11731:
11729:
11726:
11725:
11723:
11719:
11713:
11710:
11708:
11705:
11703:
11700:
11698:
11695:
11694:
11692:
11688:
11685:
11681:
11671:
11668:
11666:
11663:
11661:
11658:
11657:
11655:
11651:
11645:
11642:
11640:
11637:
11635:
11632:
11630:
11627:
11626:
11624:
11620:
11617:
11613:
11607:
11604:
11602:
11599:
11597:
11594:
11593:
11591:
11587:
11582:
11566:
11563:
11562:
11561:
11558:
11556:
11553:
11551:
11548:
11546:
11543:
11541:
11538:
11536:
11533:
11531:
11528:
11526:
11523:
11521:
11518:
11516:
11513:
11512:
11510:
11506:
11499:
11493:
11490:
11488:
11485:
11483:
11482:Powers of two
11480:
11478:
11475:
11473:
11470:
11468:
11467:Square number
11465:
11463:
11460:
11458:
11455:
11453:
11450:
11449:
11447:
11443:
11440:
11438:
11434:
11430:
11426:
11419:
11414:
11412:
11407:
11405:
11400:
11399:
11396:
11387:
11386:
11381:
11378:
11373:
11372:
11359:
11355:
11351:
11347:
11343:
11339:
11335:
11331:
11330:
11322:
11314:
11310:
11306:
11302:
11298:
11294:
11293:
11285:
11277:
11273:
11269:
11265:
11261:
11257:
11253:
11249:
11244:
11239:
11235:
11231:
11224:
11210:on 2011-06-08
11206:
11202:
11198:
11194:
11190:
11186:
11182:
11181:
11173:
11166:
11158:
11154:
11150:
11146:
11142:
11138:
11134:
11130:
11127:-functions".
11114:
11106:
11100:
11092:
11088:
11084:
11080:
11075:
11070:
11066:
11062:
11058:
11054:
11053:
11048:
11034:
11024:
11018:
11010:
11006:
11002:
10998:
10994:
10990:
10986:
10982:
10978:
10974:
10973:
10965:
10958:
10951:
10946:
10938:
10936:0-262-03384-4
10932:
10928:
10927:
10922:
10918:
10914:
10910:
10904:
10896:
10892:
10888:
10886:0-387-94415-X
10882:
10878:
10874:
10870:
10863:
10855:
10851:
10847:
10841:
10837:
10833:
10829:
10825:
10819:
10811:
10807:
10803:
10799:
10795:
10791:
10784:
10776:
10772:
10768:
10762:
10754:
10750:
10746:
10742:
10738:
10734:
10727:
10719:
10715:
10711:
10707:
10703:
10699:
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10691:
10684:
10676:
10672:
10668:
10664:
10659:
10654:
10650:
10646:
10642:
10638:
10634:
10630:
10629:Science China
10623:
10615:
10611:
10607:
10603:
10599:
10595:
10594:
10586:
10578:
10574:
10570:
10566:
10562:
10558:
10554:
10550:
10545:
10540:
10536:
10532:
10531:
10523:
10521:
10512:
10508:
10504:
10500:
10494:
10486:
10482:
10478:
10474:
10470:
10466:
10462:
10458:
10457:
10452:
10448:
10444:
10440:
10436:
10430:
10422:
10418:
10411:
10404:
10402:
10393:
10387:
10383:
10379:
10378:
10373:
10369:
10365:
10359:
10351:
10347:
10343:
10339:
10335:
10331:
10330:
10325:
10319:
10310:
10306:
10302:
10298:
10294:
10290:
10286:
10282:
10281:
10276:
10272:
10265:
10263:
10254:
10250:
10246:
10242:
10238:
10234:
10227:
10219:
10215:
10211:
10207:
10203:
10199:
10195:
10191:
10190:
10182:
10174:
10170:
10165:
10160:
10156:
10152:
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10126:
10115:
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10098:
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10072:
10066:
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10055:
10049:
10045:
10041:
10034:
10032:
10030:
10021:
10017:
10013:
10009:
10005:
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9969:
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9961:
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9951:
9950:
9945:
9939:
9937:
9929:
9925:
9921:
9917:
9913:
9909:
9902:
9896:" by Kifowit.
9895:
9889:
9885:
9884:AMATYC Review
9878:
9871:
9869:
9867:
9865:
9843:
9839:
9818:
9796:
9792:
9780:
9779:
9774:
9768:
9766:
9746:
9743:
9740:
9737:
9731:
9728:
9725:
9721:
9709:
9706:
9703:
9699:
9695:
9686:
9683:
9680:
9674:
9670:
9658:
9655:
9652:
9648:
9644:
9619:
9616:
9613:
9607:
9603:
9596:
9591:
9588:
9585:
9581:
9570:
9567:
9564:
9560:
9556:
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9544:
9541:
9535:
9531:
9519:
9516:
9513:
9509:
9498:
9495:
9492:
9488:
9484:
9479:
9476:
9464:
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9454:
9450:
9447:
9427:
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9415:
9412:
9403:
9400:
9397:
9390:
9385:
9376:
9373:
9370:
9361:
9358:
9355:
9348:
9343:
9334:
9331:
9328:
9322:
9318:
9313:
9294:
9281:
9278:
9275:
9271:
9266:
9260:
9257:
9254:
9250:
9238:
9225:
9222:
9219:
9215:
9210:
9204:
9201:
9198:
9194:
9182:
9169:
9166:
9163:
9159:
9154:
9149:
9146:
9134:
9129:
9126:
9101:
9098:
9085:
9081:
9079:
9073:
9067:
9059:
9055:
9051:
9047:
9043:
9039:
9038:
9030:
9028:
9006:
9003:
8997:
8994:
8988:
8982:
8979:
8973:
8967:
8964:
8958:
8952:
8949:
8943:
8937:
8934:
8923:
8920:
8903:
8900:
8894:
8888:
8885:
8879:
8873:
8870:
8864:
8858:
8855:
8849:
8843:
8840:
8827:
8826:
8821:
8817:
8816:
8811:
8805:
8797:
8793:
8789:
8783:
8764:
8744:
8741:
8738:
8735:
8732:
8729:
8723:
8720:
8714:
8708:
8705:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8675:
8672:
8666:
8660:
8657:
8651:
8645:
8642:
8636:
8633:
8630:
8627:
8604:
8601:
8595:
8588:
8585:
8582:
8578:
8572:
8566:
8563:
8557:
8550:
8547:
8544:
8540:
8517:
8514:
8511:
8491:
8488:
8479:
8476:
8470:
8464:
8461:
8455:
8449:
8446:
8437:
8428:
8425:
8419:
8413:
8410:
8404:
8398:
8395:
8386:
8383:
8380:
8377:
8357:
8347:
8343:
8339:
8335:
8329:
8321:
8317:
8313:
8311:0-309-08549-7
8307:
8303:
8302:
8297:
8291:
8283:
8279:
8275:
8269:
8265:
8261:
8257:
8253:
8247:
8239:
8235:
8231:
8225:
8217:
8211:
8207:
8203:
8197:
8189:
8185:
8181:
8177:
8173:
8169:
8168:
8160:
8158:
8156:
8154:
8145:
8139:
8135:
8131:
8124:
8122:
8117:
8109:
8107:
8085:
8075:
8058:
8055:
8051:
8047:
8044:
8023:
8020:
8014:
8011:
7986:
7983:
7959:
7956:
7933:
7909:
7889:
7886:
7861:
7858:
7845:
7841:
7837:
7833:
7829:
7826:It converges
7808:
7805:
7780:
7777:
7757:
7754:
7746:
7728:
7724:
7703:
7698:
7693:
7689:
7676:
7673:
7670:
7666:
7651:
7649:
7628:
7625:
7612:
7608:
7587:
7581:
7573:
7569:
7568:Basel problem
7551:
7547:
7541:
7537:
7533:
7527:
7521:
7512:
7494:
7491:
7488:
7479:
7475:
7471:
7455:
7452:
7449:
7429:
7426:
7423:
7416:
7412:
7408:
7403:
7396:
7392:
7388:
7383:
7376:
7372:
7368:
7363:
7356:
7352:
7348:
7336:
7333:
7330:
7326:
7322:
7316:
7310:
7290:
7287:
7284:
7276:
7270:
7260:
7247:
7242:
7239:
7234:
7231:
7228:
7223:
7220:
7215:
7210:
7207:
7202:
7197:
7194:
7189:
7186:
7183:
7177:
7174:
7171:
7168:
7161:
7153:
7150:
7134:
7131:
7128:
7124:
7116:
7108:
7090:
7087:
7083:
7076:
7073:
7067:
7064:
7060:
7055:
7052:
7049:
7046:
7043:
7038:
7034:
7030:
7025:
7022:
7018:
7014:
7008:
7005:
7001:
6996:
6990:
6987:
6984:
6981:
6977:
6972:
6969:
6966:
6961:
6958:
6953:
6948:
6945:
6935:
6932:
6930:
6926:
6922:
6918:
6914:
6898:
6895:
6890:
6887:
6882:
6877:
6874:
6869:
6864:
6861:
6856:
6851:
6848:
6843:
6840:
6837:
6832:
6826:
6823:
6820:
6812:
6809:
6793:
6790:
6787:
6783:
6769:
6764:
6753:
6743:
6741:
6737:
6721:
6715:
6712:
6709:
6706:
6700:
6697:
6692:
6688:
6684:
6679:
6676:
6673:
6668:
6665:
6662:
6658:
6654:
6648:
6645:
6642:
6638:
6631:
6628:
6625:
6620:
6617:
6614:
6610:
6604:
6599:
6596:
6593:
6589:
6564:
6561:
6558:
6554:
6531:
6528:
6525:
6505:
6485:
6465:
6445:
6425:
6405:
6385:
6365:
6356:
6355:expected time
6352:
6347:
6338:
6333:
6323:
6307:
6284:
6280:
6255:
6251:
6247:
6222:
6202:
6180:
6176:
6172:
6150:
6146:
6142:
6122:
6102:
6094:
6093:random graphs
6090:
6086:
6081:
6077:
6076:trading cards
6068:
6063:
6053:
6051:
6047:
6026:
6020:
6016:
6010:
6007:
6004:
6001:
5998:
5995:
5992:
5989:
5986:
5983:
5975:
5956:
5953:
5930:
5925:
5921:
5917:
5912:
5909:
5902:
5897:
5894:
5891:
5887:
5881:
5878:
5873:
5869:
5864:
5861:
5856:
5850:
5845:
5842:
5839:
5835:
5829:
5826:
5816:
5812:
5811:average order
5796:
5788:
5784:
5783:number theory
5779:
5777:
5773:
5769:
5765:
5748:
5740:
5739:Taylor series
5724:
5721:
5718:
5713:
5710:
5698:
5695:
5691:
5687:
5683:
5679:
5676:
5668:
5664:
5660:
5656:
5651:
5643:
5639:
5635:
5631:
5626:
5621:
5618:
5612:
5601:
5598:
5594:
5590:
5584:
5580:
5576:
5573:
5570:
5566:
5561:
5558:
5548:
5545:
5541:
5537:
5531:
5527:
5523:
5520:
5517:
5513:
5501:
5498:
5494:
5490:
5487:
5479:
5475:
5471:
5433:
5427:
5423:
5419:
5416:
5413:
5409:
5397:
5394:
5390:
5386:
5382:
5378:
5375:
5368:
5364:
5360:
5355:
5350:
5347:
5342:
5339:
5335:
5324:
5321:
5317:
5313:
5308:
5305:
5293:
5290:
5287:
5283:
5274:
5270:
5269:Euler product
5266:
5262:
5256:
5246:
5229:
5206:
5202:
5179:
5175:
5168:
5165:
5142:
5119:
5116:
5110:
5104:
5101:
5090:
5071:
5068:
5062:
5056:
5053:
5027:
5024:
5001:
4993:
4985:
4980:
4975:
4965:
4950:
4947:
4938:
4935:
4929:
4923:
4920:
4914:
4908:
4905:
4889:
4886:
4863:
4860:
4857:
4837:
4834:
4831:
4822:
4806:
4783:
4779:
4758:
4753:
4749:
4743:
4740:
4735:
4732:
4729:
4720:
4717:
4714:
4708:
4704:
4699:
4690:
4687:
4684:
4678:
4674:
4669:
4663:
4660:
4656:
4634:
4613:
4593:
4590:
4580:
4577:
4574:
4568:
4564:
4558:
4548:
4545:
4542:
4536:
4532:
4526:
4519:
4516:
4512:
4489:
4481:
4480:
4475:
4471:
4451:
4448:
4445:
4436:
4431:
4421:
4413:
4397:
4394:
4389:
4386:
4383:
4379:
4375:
4369:
4363:
4354:
4350:
4349:interpolation
4334:
4325:
4311:
4304:
4294:
4281:
4265:
4262:
4256:
4253:
4249:
4244:
4238:
4232:
4225:
4221:
4217:
4209:
4204:
4198:Interpolation
4195:
4193:
4177:
4174:
4169:
4165:
4144:
4141:
4136:
4132:
4111:
4108:
4103:
4099:
4090:
4074:
4070:
4066:
4059:greater than
4058:
4057:prime numbers
4040:
4036:
4026:
4011:
3990:
3987:
3984:
3958:
3954:
3949:
3945:
3921:
3917:
3913:
3909:
3900:
3895:
3892:
3889:
3885:
3881:
3876:
3872:
3849:
3845:
3822:
3812:
3796:
3774:
3750:
3746:
3738:
3720:
3716:
3693:
3690:
3685:
3681:
3665:
3651:
3629:
3625:
3621:
3617:
3613:
3610:
3605:
3601:
3597:
3594:
3586:
3570:
3567:
3564:
3542:
3538:
3534:
3528:
3525:
3521:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3493:
3489:
3480:
3476:
3466:
3453:
3448:
3445:
3438:
3433:
3430:
3427:
3423:
3419:
3414:
3410:
3386:
3382:
3371:
3367:
3351:
3329:
3326:
3323:
3320:
3319:
3307:
3304:
3301:
3298:
3297:
3285:
3282:
3279:
3276:
3275:
3263:
3260:
3257:
3254:
3253:
3241:
3238:
3235:
3232:
3231:
3219:
3216:
3213:
3210:
3209:
3197:
3194:
3191:
3188:
3187:
3175:
3172:
3169:
3166:
3165:
3153:
3150:
3147:
3144:
3143:
3131:
3128:
3125:
3122:
3121:
3109:
3106:
3103:
3100:
3099:
3087:
3084:
3081:
3078:
3077:
3065:
3062:
3059:
3056:
3055:
3043:
3040:
3037:
3034:
3033:
3021:
3018:
3015:
3012:
3011:
2999:
2996:
2993:
2990:
2989:
2977:
2974:
2971:
2968:
2967:
2955:
2952:
2949:
2946:
2945:
2933:
2930:
2927:
2924:
2923:
2908:
2902:
2901:
2897:
2894:
2890:
2873:
2869:
2846:
2838:
2834:
2824:
2822:
2805:
2783:
2780:
2777:
2774:
2768:
2765:
2758:
2753:
2749:
2745:
2740:
2737:
2730:
2725:
2722:
2719:
2715:
2711:
2708:
2705:
2699:
2696:
2689:
2686:
2683:
2678:
2674:
2663:
2649:
2643:
2640:
2637:
2631:
2628:
2616:
2612:
2603:
2587:
2564:
2561:
2555:
2552:
2529:
2526:
2515:
2495:
2491:
2487:
2484:
2481:
2472:
2466:Integral test
2463:
2461:
2457:
2456:Nicole Oresme
2439:
2436:
2431:
2428:
2425:
2420:
2417:
2408:
2404:
2398:
2395:
2392:
2388:
2366:
2359:
2355:
2351:
2330:
2327:
2324:
2319:
2316:
2311:
2306:
2303:
2298:
2293:
2290:
2285:
2280:
2277:
2272:
2269:
2262:
2253:
2250:
2246:
2240:
2237:
2232:
2229:
2226:
2221:
2218:
2212:
2208:
2204:
2198:
2195:
2190:
2185:
2182:
2177:
2172:
2169:
2164:
2159:
2156:
2150:
2146:
2142:
2136:
2133:
2128:
2123:
2120:
2114:
2110:
2106:
2101:
2098:
2093:
2089:
2086:
2057:
2054:
2052:
2035:
2030:
2028:
2020:
2017:
2012:
2010:
1993:
1988:
1986:
1969:
1964:
1962:
1945:
1940:
1938:
1930:
1927:
1922:
1920:
1903:
1898:
1896:
1888:
1885:
1880:
1878:
1873:
1870:
1861:
1858:
1856:
1848:
1845:
1840:
1838:
1830:
1827:
1822:
1820:
1812:
1809:
1804:
1802:
1794:
1791:
1786:
1784:
1776:
1773:
1768:
1766:
1758:
1755:
1750:
1748:
1740:
1737:
1732:
1730:
1722:
1719:
1714:
1712:
1707:
1695:
1686:
1677:
1674:
1670:
1666:
1650:
1647:
1642:
1639:
1634:
1629:
1626:
1621:
1616:
1613:
1608:
1603:
1600:
1595:
1592:
1589:
1584:
1581:
1569:
1566:
1563:
1559:
1544:
1542:
1539:, in 1968 by
1524:
1520:
1511:
1506:
1504:
1500:
1496:
1492:
1488:
1484:
1483:Nicole Oresme
1479:
1477:
1473:
1469:
1465:
1461:
1457:
1456:harmonic mean
1453:
1449:
1448:harmonic mean
1445:
1424:
1421:
1393:
1390:
1362:
1359:
1347:
1343:
1340:or harmonics
1339:
1320:
1317:
1311:
1308:
1302:
1296:
1293:
1287:
1284:
1275:
1266:
1264:
1260:
1256:
1252:
1248:
1247:random graphs
1244:
1240:
1236:
1231:
1229:
1225:
1221:
1217:
1216:Nicole Oresme
1213:
1209:
1193:
1190:
1187:
1179:
1163:
1143:
1140:
1137:
1134:
1131:
1111:
1102:
1089:
1086:
1083:
1078:
1075:
1070:
1065:
1062:
1057:
1052:
1049:
1044:
1039:
1036:
1031:
1028:
1025:
1020:
1017:
1005:
1002:
999:
995:
986:
982:
978:
974:
962:
957:
955:
950:
948:
943:
942:
940:
939:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
898:
890:
889:
882:
879:
877:
874:
872:
869:
867:
864:
863:
855:
854:
843:
840:
838:
835:
833:
830:
829:
828:
827:
817:
816:
805:
802:
800:
797:
795:
792:
790:
787:
785:
784:Line integral
782:
780:
777:
775:
772:
771:
770:
769:
765:
764:
759:
756:
754:
751:
749:
746:
744:
741:
740:
739:
738:
734:
733:
727:
726:Multivariable
721:
720:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
680:
679:
678:
674:
673:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
639:
638:
637:
631:
625:
624:
613:
610:
608:
605:
603:
600:
598:
595:
593:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
563:
562:
561:
558:
555:
554:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
523:
519:
516:
515:
514:
513:
507:
501:
500:
489:
486:
484:
481:
479:
476:
474:
471:
468:
464:
461:
459:
456:
453:
449:
445:
444:trigonometric
441:
438:
436:
433:
431:
428:
426:
423:
422:
421:
420:
416:
415:
410:
407:
405:
402:
400:
397:
395:
392:
389:
385:
382:
380:
377:
376:
375:
374:
370:
369:
364:
361:
359:
356:
354:
351:
350:
349:
348:
342:
336:
335:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
275:
274:
273:
270:
267:
266:
261:
258:
256:
255:Related rates
253:
251:
248:
246:
243:
241:
238:
236:
233:
232:
231:
230:
226:
225:
218:
215:
213:
212:of a function
210:
208:
207:infinitesimal
205:
204:
203:
200:
197:
193:
190:
189:
188:
187:
183:
182:
176:
170:
169:
163:
160:
158:
155:
153:
150:
149:
144:
141:
139:
136:
135:
131:
128:
127:
126:
125:
106:
100:
97:
91:
85:
82:
79:
76:
69:
62:
59:
53:
48:
44:
35:
34:
31:
28:
27:
23:
22:
16:
11798:Power series
11767:
11540:Lucas number
11492:Powers of 10
11472:Cubic number
11383:
11333:
11327:
11321:
11296:
11290:
11284:
11233:
11229:
11223:
11212:. Retrieved
11205:the original
11184:
11178:
11165:
11135:(1): 11–59.
11132:
11128:
11105:Bombieri, E.
11099:
11056:
11050:
11017:
10976:
10970:
10957:
10945:
10925:
10903:
10868:
10862:
10827:
10824:Frieze, Alan
10818:
10793:
10789:
10783:
10774:
10767:Parker, Matt
10761:
10739:(2): 89–93.
10736:
10732:
10726:
10693:
10689:
10683:
10658:10722/129254
10632:
10628:
10622:
10600:(1): 13–20.
10597:
10591:
10585:
10534:
10528:
10510:
10509:(in Latin).
10506:
10493:
10460:
10454:
10439:Peres, Yuval
10429:
10420:
10416:
10376:
10358:
10333:
10327:
10318:
10284:
10278:
10274:
10236:
10232:
10226:
10193:
10187:
10181:
10164:2318/1622121
10146:
10140:
10114:
10080:
10074:
10043:
9995:
9989:
9976:
9948:
9927:
9915:
9911:
9907:
9901:
9887:
9883:
9777:
9083:
9077:
9066:
9044:(1): 18–23.
9041:
9035:
8829:
8819:
8814:
8804:
8795:
8791:
8782:
8504:. Since for
8345:
8341:
8328:
8300:
8290:
8255:
8246:
8237:
8233:
8224:
8205:
8196:
8171:
8165:
8133:
8087:
7795:probability
7657:
7272:
7109:
6936:
6933:
6912:
6774:
6343:
6089:connectivity
6073:
5780:
5273:prime number
5258:
4989:
4823:
4477:
4470:jeep problem
4467:
4430:Jeep problem
4419:
4416:Applications
4213:
4027:
3737:power of two
3671:
3668:Divisibility
3472:
3343:
2827:Partial sums
2664:
2511:
1694:power of two
1691:
1673:partial sums
1550:
1541:Donald Knuth
1507:
1480:
1459:
1455:
1335:
1255:cantilevered
1232:
1103:
976:
970:
527:
440:Substitution
202:Differential
175:Differential
15:
11665:Conditional
11653:Convergence
11644:Telescoping
11629:Alternating
11545:Pell number
11260:2434/634047
10324:Gale, David
9912:SIAM Review
9084:Opera Omnia
8232:(c. 1360).
7793:with equal
7607:Roger Apéry
7511:simple pole
6775:The series
6740:linear time
4647:th trip is
4087:, and uses
3469:Growth rate
3368:, called a
3366:partial sum
1472:floor plans
1468:proportions
1346:wavelengths
1265:algorithm.
973:mathematics
901:Precalculus
894:Miscellanea
859:Specialized
766:Definitions
533:Alternating
371:Definitions
184:Definitions
11690:Convergent
11634:Convergent
11243:1806.05402
11214:2006-08-07
10777:. YouTube.
10544:1701.04718
10513:: 160–188.
10451:Zwick, Uri
10239:: 89–108.
8112:References
7615:real part
7605:proved by
7442:which for
6923:, but not
6761:See also:
6750:See also:
6087:, and the
5265:formal sum
4353:factorials
1667:. It is a
1476:elevations
1257:, and the
1104:The first
881:Variations
876:Stochastic
866:Fractional
735:Formalisms
698:Divergence
667:Identities
647:Divergence
192:Derivative
143:Continuity
11721:Divergent
11639:Divergent
11501:Advanced
11477:Factorial
11425:Sequences
11385:MathWorld
11276:119160796
11157:120058240
11091:202575422
11035:π
11023:Soddy, F.
10810:109194745
10753:119887116
10718:126381029
10577:119165483
10309:125835186
10253:254990799
10149:: 41–46.
10105:124359670
9744:−
9715:∞
9700:∑
9664:∞
9649:∑
9582:∑
9576:∞
9561:∑
9525:∞
9510:∑
9504:∞
9489:∑
9470:∞
9455:∑
9428:⋯
9295:⋯
9267:−
9211:−
9155:−
9007:⋯
8733:⋯
8685:⋯
8548:−
8492:⋯
8056:−
8045:ε
8024:ε
8021:−
7984:±
7957:−
7887:−
7848:close to
7778:−
7682:∞
7667:∑
7609:to be an
7582:ζ
7538:π
7522:ζ
7427:⋯
7342:∞
7327:∑
7311:ζ
7240:π
7232:⋯
7216:−
7190:−
7151:−
7140:∞
7125:∑
7088:−
7056:−
7050:
7031:−
6997:−
6988:−
6970:⋯
6954:−
6899:⋯
6896:−
6870:−
6844:−
6810:−
6799:∞
6784:∑
6713:
6677:−
6659:∑
6629:−
6611:∑
6590:∑
6346:quicksort
6332:Quicksort
6236:down to 1
6002:−
5999:γ
5987:
5888:∑
5874:≤
5836:∑
5699:∈
5692:∑
5680:⋯
5602:∈
5595:∑
5574:−
5562:
5549:∈
5542:∑
5521:−
5502:∈
5495:∏
5491:
5417:−
5398:∈
5391:∏
5379:⋯
5325:∈
5318:∏
5299:∞
5284:∑
5259:In 1737,
5111:⋅
5063:⋅
4733:⋯
4718:−
4688:−
4594:…
4578:−
4546:−
4398:γ
4395:−
4387:−
4364:ψ
4320:Γ
4302:Γ
4276:Γ
4266:
4233:ψ
3886:∑
3611:≤
3602:ε
3598:≤
3568:≈
3565:γ
3539:ε
3535:−
3514:γ
3505:
3424:∑
3330:~3.59774
3327:/15519504
3308:~3.54774
3305:/77597520
3302:275295799
3286:~3.49511
3264:~3.43955
3261:/12252240
3242:~3.38073
3220:~3.31823
3198:~3.25156
3176:~3.18013
3154:~3.10321
3132:~3.01988
3110:~2.92897
3088:~2.82897
3066:~2.71786
3044:~2.59286
3000:~2.28333
2978:~2.08333
2956:~1.83333
2750:∫
2716:∑
2675:∫
2647:∞
2622:∞
2613:∫
2426:≥
2389:∑
2328:⋯
2254:⋯
2230:⋯
2058:⋯
1871:≥
1862:⋯
1651:⋯
1575:∞
1560:∑
1338:overtones
1321:…
1263:quicksort
1191:≈
1188:γ
1144:γ
1135:
1087:⋯
1011:∞
996:∑
871:Malliavin
758:Geometric
657:Laplacian
607:Dirichlet
518:Geometric
98:−
45:∫
11909:Category
11894:Category
11660:Absolute
11358:11461182
11350:27642532
11025:(1943).
11009:20575373
10501:(1737).
10097:24496876
9946:(2007).
9924:20454048
9074:(1742).
8812:(1713).
8790:(1689).
8336:(1650).
8298:(2003).
8254:(2010).
8204:(2001).
8132:(eds.).
8090:22.92067
7947:or less
6915:. It is
5870:⌋
5857:⌊
5787:divisors
4305:′
3374:denoted
3324:55835135
3283:/4084080
3280:14274301
3258:42142223
2354:positive
1342:in music
1224:integral
1156:, where
911:Glossary
821:Advanced
799:Jacobian
753:Exterior
683:Gradient
675:Theorems
642:Gradient
581:Integral
543:Binomial
528:Harmonic
388:improper
384:Integral
341:Integral
323:Reynolds
298:Quotient
227:Concepts
63:′
30:Calculus
11670:Uniform
11313:2321096
11268:3907571
11201:3647827
11149:2684771
11083:0009207
11061:Bibcode
11001:2663251
10895:1329545
10854:3675279
10710:3607889
10675:6168120
10667:2718848
10637:Bibcode
10614:3300350
10569:3738242
10485:1713091
10477:2572086
10350:2317382
10301:3620384
10218:1572267
10210:2689999
10173:3486261
10020:0289994
10012:2316476
9968:2284828
9058:2686312
8620:, then
8320:1968857
8282:2667826
8188:2687471
7976:values
7481:except
7476:on all
6919:by the
6299:is the
6080:parkrun
5813:of the
5245:layer.
5221:is the
4990:In the
4798:is the
4351:of the
4222:of the
3809:is the
3583:is the
3334:3.59774
3312:3.54774
3290:3.49511
3268:3.43955
3246:3.38073
3239:/720720
3236:2436559
3224:3.31823
3217:/360360
3214:1195757
3202:3.25156
3195:/360360
3192:1171733
3180:3.18013
3173:/360360
3170:1145993
3158:3.10321
3136:3.01988
3114:2.92897
3092:2.82897
3070:2.71786
3048:2.59286
3004:2.28333
2982:2.08333
2960:1.83333
2895:decimal
2358:integer
1497:and by
1464:Baroque
1269:History
1261:of the
1206:is the
1176:is the
979:is the
906:History
804:Hessian
693:Stokes'
688:Green's
520: (
442: (
386: (
308:Inverse
283:Product
194: (
11622:Series
11429:series
11356:
11348:
11311:
11274:
11266:
11199:
11155:
11147:
11089:
11081:
11007:
10999:
10991:
10933:
10893:
10883:
10852:
10842:
10808:
10751:
10716:
10708:
10673:
10665:
10612:
10575:
10567:
10559:
10483:
10475:
10388:
10348:
10307:
10299:
10251:
10216:
10208:
10171:
10103:
10095:
10050:
10018:
10010:
9966:
9956:
9922:
9056:
8318:
8308:
8280:
8270:
8212:
8186:
8140:
7842:whose
6272:where
5446:where
4771:where
4474:Alcuin
4157:, and
3977:(when
3571:0.5772
3557:where
3337:
3315:
3293:
3271:
3249:
3227:
3205:
3183:
3161:
3151:/27720
3139:
3129:/27720
3117:
3095:
3073:
3051:
3029:
3007:
2985:
2963:
2941:
2919:
1344:: the
1249:, the
975:, the
748:Tensor
743:Matrix
630:Vector
548:Taylor
506:Series
138:Limits
11565:array
11445:Basic
11354:S2CID
11346:JSTOR
11309:JSTOR
11272:S2CID
11238:arXiv
11208:(PDF)
11197:JSTOR
11175:(PDF)
11153:S2CID
11087:S2CID
11005:S2CID
10989:JSTOR
10967:(PDF)
10806:S2CID
10749:S2CID
10714:S2CID
10706:JSTOR
10671:S2CID
10573:S2CID
10557:JSTOR
10539:arXiv
10481:S2CID
10413:(PDF)
10346:JSTOR
10305:S2CID
10297:JSTOR
10249:S2CID
10206:JSTOR
10101:S2CID
10093:JSTOR
10008:JSTOR
9920:JSTOR
9880:(PDF)
9054:JSTOR
8818:[
8794:[
8344:[
8236:[
8184:JSTOR
8098:34816
8095:64150
8092:66192
7949:than
7926:than
7472:to a
6215:from
3837:then
3815:1 to
3767:1 to
3148:86021
3126:83711
3107:/2520
3085:/2520
3022:2.45
1474:, of
1194:0.577
571:Ratio
538:Power
452:Euler
430:Discs
425:Parts
293:Power
288:Chain
217:total
11505:list
11427:and
10931:ISBN
10881:ISBN
10840:ISBN
10386:ISBN
10048:ISBN
9954:ISBN
9117:as
8631:>
8596:>
8515:>
8306:ISBN
8268:ISBN
8210:ISBN
8138:ISBN
8106:base
8048:<
7902:and
7770:and
7743:are
7288:>
7273:The
6438:and
6378:and
6344:The
4982:The
4951:27.5
4468:The
4214:The
4206:The
4178:2.45
3988:>
3587:and
3372:and
3104:7381
3082:7129
3063:/280
3041:/140
3026:2.45
2934:1.5
2746:<
2712:<
1458:and
1180:and
652:Curl
612:Abel
576:Root
11338:doi
11334:115
11301:doi
11256:hdl
11248:doi
11234:356
11189:doi
11185:110
11137:doi
11069:doi
11057:182
10981:doi
10977:117
10873:doi
10832:doi
10798:doi
10741:doi
10698:doi
10653:hdl
10645:doi
10602:doi
10549:doi
10465:doi
10461:116
10338:doi
10289:doi
10277:".
10241:doi
10198:doi
10159:hdl
10151:doi
10147:166
10085:doi
10000:doi
9046:doi
8260:doi
8176:doi
8101:...
7846:is
6710:log
6478:or
6240:is
6165:is
6091:of
4145:1.5
3789:If
3060:761
3038:363
3019:/20
2997:/60
2994:137
2975:/12
2938:1.5
2798:of
1470:of
1245:of
971:In
278:Sum
11911::
11382:.
11352:.
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11332:.
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11297:86
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8381:=
8378:S
8358:S
8322:.
8284:.
8262::
8218:.
8190:.
8178::
8146:.
8073:.
8015:8
8012:1
7999:,
7987:2
7972:.
7960:3
7934:3
7922:,
7910:1
7890:1
7862:4
7859:1
7824:.
7809:2
7806:1
7781:1
7758:1
7755:+
7729:n
7725:s
7704:,
7699:n
7694:n
7690:s
7677:1
7674:=
7671:n
7644:,
7629:2
7626:1
7603:,
7591:)
7588:3
7585:(
7564:,
7552:6
7548:/
7542:2
7534:=
7531:)
7528:2
7525:(
7507:,
7495:1
7492:=
7489:x
7456:1
7453:=
7450:x
7430:,
7424:+
7417:x
7413:3
7409:1
7404:+
7397:x
7393:2
7389:1
7384:+
7377:x
7373:1
7369:1
7364:=
7357:x
7353:n
7349:1
7337:1
7334:=
7331:n
7323:=
7320:)
7317:x
7314:(
7291:1
7285:x
7248:.
7243:4
7235:=
7229:+
7224:7
7221:1
7211:5
7208:1
7203:+
7198:3
7195:1
7187:1
7184:=
7178:1
7175:+
7172:n
7169:2
7162:n
7158:)
7154:1
7148:(
7135:0
7132:=
7129:n
7114:π
7096:)
7091:2
7084:n
7080:(
7077:O
7074:+
7068:n
7065:2
7061:1
7053:2
7044:=
7039:n
7035:H
7026:n
7023:2
7019:H
7015:=
7009:n
7006:2
7002:1
6991:1
6985:n
6982:2
6978:1
6973:+
6967:+
6962:2
6959:1
6949:1
6946:1
6891:5
6888:1
6883:+
6878:4
6875:1
6865:3
6862:1
6857:+
6852:2
6849:1
6841:1
6838:=
6833:n
6827:1
6824:+
6821:n
6817:)
6813:1
6807:(
6794:1
6791:=
6788:n
6722:.
6719:)
6716:n
6707:n
6704:(
6701:O
6698:=
6693:i
6689:H
6685:2
6680:1
6674:n
6669:1
6666:=
6663:i
6655:=
6649:2
6646:+
6643:k
6639:2
6632:2
6626:i
6621:0
6618:=
6615:k
6605:n
6600:2
6597:=
6594:i
6565:2
6562:+
6559:k
6555:2
6532:2
6529:+
6526:k
6506:k
6486:y
6466:x
6446:y
6426:x
6406:k
6386:y
6366:x
6308:n
6285:n
6281:H
6270:,
6256:n
6252:H
6248:n
6223:n
6203:k
6193:.
6181:k
6177:/
6173:n
6151:n
6147:/
6143:k
6123:n
6103:k
6032:)
6027:n
6021:/
6017:1
6014:(
6011:O
6008:+
6005:1
5996:2
5993:+
5990:n
5957:n
5954:1
5931:.
5926:n
5922:H
5918:=
5913:i
5910:n
5903:n
5898:1
5895:=
5892:i
5882:n
5879:1
5865:i
5862:n
5851:n
5846:1
5843:=
5840:i
5830:n
5827:1
5797:n
5749:K
5725:.
5722:K
5719:+
5714:p
5711:1
5703:P
5696:p
5688:=
5684:)
5677:+
5669:3
5665:p
5661:3
5657:1
5652:+
5644:2
5640:p
5636:2
5632:1
5627:+
5622:p
5619:1
5613:(
5606:P
5599:p
5591:=
5585:p
5581:/
5577:1
5571:1
5567:1
5553:P
5546:p
5538:=
5532:p
5528:/
5524:1
5518:1
5514:1
5506:P
5499:p
5455:P
5434:,
5428:p
5424:/
5420:1
5414:1
5410:1
5402:P
5395:p
5387:=
5383:)
5376:+
5369:2
5365:p
5361:1
5356:+
5351:p
5348:1
5343:+
5340:1
5336:(
5329:P
5322:p
5314:=
5309:i
5306:1
5294:1
5291:=
5288:i
5230:n
5207:n
5203:H
5180:n
5176:H
5169:2
5166:1
5143:n
5120:3
5117:1
5105:2
5102:1
5072:2
5069:1
5057:2
5054:1
5028:2
5025:1
5002:n
4948:=
4945:)
4939:1
4936:1
4930:+
4924:2
4921:1
4915:+
4909:3
4906:1
4898:(
4890:2
4864:3
4861:=
4858:n
4835:=
4832:r
4807:n
4784:n
4780:H
4759:,
4754:n
4750:H
4744:2
4741:r
4736:=
4730:+
4724:)
4721:2
4715:n
4712:(
4709:2
4705:r
4700:+
4694:)
4691:1
4685:n
4682:(
4679:2
4675:r
4670:+
4664:n
4661:2
4657:r
4635:n
4614:r
4591:,
4584:)
4581:2
4575:n
4572:(
4569:2
4565:r
4559:,
4552:)
4549:1
4543:n
4540:(
4537:2
4533:r
4527:,
4520:n
4517:2
4513:r
4490:n
4452:3
4449:=
4446:n
4410:.
4390:1
4384:n
4380:H
4376:=
4373:)
4370:n
4367:(
4335:.
4329:)
4326:x
4323:(
4315:)
4312:x
4309:(
4295:=
4290:)
4285:)
4282:x
4279:(
4271:(
4257:x
4254:d
4250:d
4245:=
4242:)
4239:x
4236:(
4175:=
4170:6
4166:H
4142:=
4137:2
4133:H
4112:1
4109:=
4104:1
4100:H
4075:2
4071:/
4067:n
4041:n
4037:H
4012:M
4003:)
3991:1
3985:n
3973:,
3959:k
3955:2
3950:/
3946:M
3922:M
3918:i
3914:/
3910:M
3901:n
3896:1
3893:=
3890:i
3882:=
3877:n
3873:H
3850:k
3846:H
3835:,
3823:n
3797:M
3787:.
3775:n
3751:k
3747:2
3721:n
3717:H
3706:.
3694:1
3691:=
3686:1
3682:H
3652:n
3630:2
3626:n
3622:8
3618:/
3614:1
3606:n
3595:0
3543:n
3529:n
3526:2
3522:1
3517:+
3511:+
3508:n
3499:=
3494:n
3490:H
3454:.
3449:k
3446:1
3439:n
3434:1
3431:=
3428:k
3420:=
3415:n
3411:H
3401::
3387:n
3383:H
3352:n
3079:9
3057:8
3035:7
3013:6
2991:5
2969:4
2947:3
2928:3
2925:2
2916:1
2910:~
2906:1
2903:1
2874:n
2870:H
2847:n
2806:n
2781:+
2778:x
2775:d
2769:x
2766:1
2759:N
2754:1
2741:i
2738:1
2731:N
2726:1
2723:=
2720:i
2709:x
2706:d
2700:x
2697:1
2690:1
2687:+
2684:N
2679:1
2650:.
2644:=
2641:x
2638:d
2632:x
2629:1
2617:1
2588:x
2565:x
2562:1
2556:=
2553:y
2530:n
2527:1
2496:x
2492:/
2488:1
2485:=
2482:y
2440:2
2437:k
2432:+
2429:1
2421:n
2418:1
2409:k
2405:2
2399:1
2396:=
2393:n
2379:,
2367:k
2331:.
2325:+
2320:2
2317:1
2312:+
2307:2
2304:1
2299:+
2294:2
2291:1
2286:+
2281:2
2278:1
2273:+
2270:1
2263:=
2251:+
2247:)
2238:1
2233:+
2227:+
2219:1
2213:(
2209:+
2205:)
2199:8
2196:1
2191:+
2186:8
2183:1
2178:+
2173:8
2170:1
2165:+
2160:8
2157:1
2151:(
2147:+
2143:)
2137:4
2134:1
2129:+
2124:4
2121:1
2115:(
2111:+
2107:)
2102:2
2099:1
2094:(
2090:+
2087:1
2055:+
2036:1
2031:+
2021:8
2018:1
2013:+
2000:8
1994:1
1989:+
1976:8
1970:1
1965:+
1952:8
1946:1
1941:+
1931:4
1928:1
1923:+
1910:4
1904:1
1899:+
1889:2
1886:1
1881:+
1874:1
1859:+
1849:9
1846:1
1841:+
1831:8
1828:1
1823:+
1813:7
1810:1
1805:+
1795:6
1792:1
1787:+
1777:5
1774:1
1769:+
1759:4
1756:1
1751:+
1741:3
1738:1
1733:+
1723:2
1720:1
1715:+
1708:1
1648:+
1643:5
1640:1
1635:+
1630:4
1627:1
1622:+
1617:3
1614:1
1609:+
1604:2
1601:1
1596:+
1593:1
1590:=
1585:n
1582:1
1570:1
1567:=
1564:n
1525:n
1521:H
1440:,
1425:4
1422:1
1409:,
1394:3
1391:1
1378:,
1363:2
1360:1
1318:,
1312:3
1309:1
1303:,
1297:2
1294:1
1288:,
1285:1
1141:+
1138:n
1112:n
1090:.
1084:+
1079:5
1076:1
1071:+
1066:4
1063:1
1058:+
1053:3
1050:1
1045:+
1040:2
1037:1
1032:+
1029:1
1026:=
1021:n
1018:1
1006:1
1003:=
1000:n
960:e
953:t
946:v
524:)
469:)
465:(
454:)
390:)
198:)
110:)
107:a
104:(
101:f
95:)
92:b
89:(
86:f
83:=
80:t
77:d
73:)
70:t
67:(
60:f
54:b
49:a
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