3032:
3459:
3341:
2071:"... problems that arise naturally (i.e., from nature) do have solutions, so the assumption that things will work out eventually is justified experimentally without the need for existence sorts of proof. Assume everything is okay, and if the arrived-at solution works, you were probably right, or at least right enough. ... so why bother with the details that only show up in homework problems?"
1946:"A posteriori, however, the explanation of this lack of shock on the part of the students may be somewhat different. They accepted calmly the absurdity because, after all, 'mathematics is completely abstract and far from reality', and 'with those mathematical transformations you can prove all kinds of nonsense', as one of the boys later said."
2063:"Since series are generally presented without history and separate from applications, the student must wonder not only "What are these things?" but also "Why are we doing this?" The preoccupation with determining convergence but not the sum makes the whole process seem artificial and pointless to many students—and instructors as well."
1895:
as a result of the geometric series formula. Ideally, by searching for the error in reasoning and by investigating the formula for various common ratios, the students would "notice that there are two kinds of series and an implicit conception of convergence will be born". However, the students showed
713:
2050:
as being the average of 0 and 1. Bagni notes that their reasoning, while similar to
Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics. He concludes that the responses are consistent with a link between historical development and individual development,
1556:
1269:
1412:
1705:. Otherwise these operations can alter the result of summation. Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1. For instance, the series
1102:
507:
1991:
The students had been introduced to the idea of an infinite set, but they had no prior experience with infinite series. They were given ten minutes without books or calculators. The 88 responses were categorized as follows:
2318:
343:
474:
2033:
The researcher, Giorgio Bagni, interviewed several of the students to determine their reasoning. Some 16 of them justified an answer of 0 using logic similar to that of Grandi and
Riccati. Others justified
2059:
Joel
Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics.
1841:
of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5.
1828:
2075:
Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Jean-Charles Callet. Euler had viewed the sum as the evaluation at
1423:
512:
877:. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:
2169:
1645:
1151:
1950:
The students were ultimately not immune to the question of convergence; Sierpińska succeeded in engaging them in the issue by linking it to decimal expansions the following day. As soon as
1833:(in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms – the infinitude of both +1s and −1s allows any finite number of 1s or −1s to be prepended, by
88:
1865:. She focused on humanities students with the expectation that their mathematical experience would be less significant than that of their peers studying mathematics and physics, so the
896:
In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century
1306:
205:
1143:
1298:
985:
819:
971:
749:
868:
2730:
Lehmann, Joel (1995). "Converging
Concepts of Series: Learning from History". In Swetz, Frank; Fauvel, John; Bekken, Otto; Johansson, Bengt; Katz, Victor (eds.).
934:
839:
3223:
772:
479:
Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". This is closely akin to the general problem of
1701:
It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is
708:{\displaystyle {\begin{aligned}S&=1-1+1-1+\ldots ,{\text{ so}}\\1-S&=1-(1-1+1-1+\ldots )=1-1+1-1+\ldots =S\\1-S&=S\\1&=2S,\end{aligned}}}
2338:. Lehman argues that seeing such a conflicting outcome in intuitive evaluations may motivate the need for rigorous definitions and attention to detail.
2400:
2201:
1876:
Sierpińska initially expected the students to balk at assigning a value to Grandi's series, at which point she could shock them by claiming that
2866:
3213:
3377:
3306:
220:
354:
3147:
2387:. This is the square of the value most summation methods assign to Grandi's series, which is reasonable as it can be viewed as the
873:
The above manipulations do not consider what the sum of a series rigorously means and how said algebraic methods can be applied to
116:
However, though it is divergent, it can be manipulated to yield a number of mathematically interesting results. For example, many
1834:
1711:
1551:{\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}
2577:
3157:
2759:
2720:
2695:
2634:
2608:
1264:{\displaystyle \sum _{n=0}^{\infty }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}
3321:
3152:
2912:
2859:
2751:
3412:
3301:
2358:
2090:
1600:
904:, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between
3370:
3311:
36:
3503:
3407:
3402:
3203:
3193:
2425:
2410:
2347:
1407:{\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}
3508:
3397:
3316:
3218:
2852:
1668:
142:
3344:
1110:
3483:
3462:
3363:
3326:
2818:
484:
502:
and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:
3208:
2823:
2712:
1277:
874:
499:
3493:
3488:
3198:
3188:
3178:
2420:
2415:
2405:
1917:
1097:{\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}
2658:
976:
495:. By taking the average of these two "values", one can justify that the series converges to 1/2.
3293:
3115:
480:
120:
are used in mathematics to assign numerical values even to a divergent series. For example, the
2955:
2902:
2653:
2189:. However, Callet pointed out that one could instead view Grandi's series as the evaluation at
348:
On the other hand, a similar bracketing procedure leads to the apparently contradictory result
2745:
3162:
2907:
1592:
941:
777:
3273:
3110:
2879:
1926:, the students' reaction shouldn't be too surprising given that Leibniz and Grandi thought
1702:
720:
2625:
Mathematics, the science of patterns: the search for order in life, mind, and the universe
844:
8:
3498:
3253:
3120:
2430:
1679:
897:
125:
2772:(November 1987). "Humanities students and epistemological obstacles related to limits".
2435:
121:
3428:
3094:
3079:
3051:
3031:
2970:
2797:
2789:
2671:
2623:
1691:
919:
824:
211:
2769:
1855:
3444:
3283:
3084:
3056:
3010:
3000:
2980:
2965:
2801:
2755:
2716:
2706:
2691:
2630:
2604:
2440:
1975:
754:
3268:
3089:
3015:
3005:
2985:
2887:
2781:
2663:
2588:
2083:
1695:
1565:
117:
110:
1957:
caught the students by surprise, the rest of her material "went past their ears".
3046:
2975:
2809:
1987:(addends, infinitely many, are always +1 and –1). What is your opinion about it?"
901:
24:
3278:
3263:
3258:
2937:
2922:
2388:
1858:
introduced Grandi's series to a group of 17-year-old precalculus students at a
2838:
3477:
3243:
2917:
2731:
2708:
The theory of functions of a real variable and the theory of
Fourier's series
905:
113:, meaning that the sequence of partial sums of the series does not converge.
98:
1979:
pupils (between 16 and 18 years old) were given cards asking the following:
3355:
3248:
2990:
2932:
2813:
2683:
2618:
2313:{\displaystyle 1-x^{2}+x^{3}-x^{5}+x^{6}-\cdots ={\tfrac {1+x}{1+x+x^{2}}}}
1866:
106:
1869:
obstacles they exhibit would be more representative of the obstacles that
1678:
In modern mathematics, the sum of an infinite series is defined to be the
2995:
2942:
1838:
1683:
488:
102:
20:
2844:
2793:
2785:
2675:
2371:
infinity) is also divergent, but some methods may be used to sum it to
1690:
which clearly does not approach any number (although it does have two
2927:
1417:
However, as mentioned, the series obtained by switching the limits,
2667:
2875:
2067:
As a result, many students develop an attitude similar to Euler's:
1952:
1686:, if it exists. The sequence of partial sums of Grandi's series is
338:{\displaystyle (1-1)+(1-1)+(1-1)+(1-1)+\ldots =0+0+0+0+\ldots =0.}
1966:
492:
469:{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+\ldots =1+0+0+0+\ldots =1.}
109:, who gave a memorable treatment of the series in 1703. It is a
2516:
2514:
2512:
1983:"In 1703, the mathematician Guido Grandi studied the addition:
1862:
1859:
2368:
1970:
2644:
Kline, Morris (November 1983). "Euler and
Infinite Series".
2509:
2585:
International
Journal for Mathematics Teaching and Learning
1823:{\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }
2739:. Mathematical Association of America. pp. 161–180.
2555:
2553:
2498:
2578:"Infinite Series from History to Mathematics Education"
2472:
2470:
2457:
2455:
2270:
1604:
780:
757:
3224:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
3214:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
2550:
2538:
2204:
2093:
1714:
1603:
1426:
1309:
1280:
1154:
1113:
988:
944:
922:
847:
827:
723:
510:
357:
223:
145:
39:
2744:Protter, Murray H.; Morrey, Charles B. Jr. (1991).
2526:
2467:
2452:
911:
2622:
2312:
2163:
1822:
1649:, which is only defined on the complex unit disk,
1639:
1550:
1406:
1292:
1263:
1137:
1096:
965:
928:
862:
833:
813:
766:
743:
707:
468:
337:
199:
82:
16:Infinite series summing alternating 1 and -1 terms
1973:around the year 2000, third-year and fourth-year
136:One obvious method to find the sum of the series
3475:
1444:
1428:
1327:
1311:
990:
2808:
2164:{\displaystyle 1-x+x^{2}-x^{3}+\cdots =1/(1+x)}
1640:{\displaystyle \textstyle \sum _{n=0}^{N}z^{n}}
751:. The same conclusion results from calculating
2839:One minus one plus one minus one – Numberphile
3371:
2860:
2743:
2504:
83:{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}}
3385:
3307:Hypergeometric function of a matrix argument
2051:although the cultural context is different.
3163:1 + 1/2 + 1/3 + ... (Riemann zeta function)
1694:at 0 and 1). Therefore, Grandi's series is
3378:
3364:
2867:
2853:
2768:
2520:
483:, and variations of this idea, called the
3219:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
2688:Theory and Application of Infinite Series
2657:
2874:
2729:
2601:Fourier Series and Orthogonal Functions
2559:
2544:
214:and perform the subtractions in place:
200:{\displaystyle 1-1+1-1+1-1+1-1+\ldots }
3476:
2704:
2617:
2461:
131:
3359:
2848:
2682:
2643:
2598:
2575:
2532:
2492:
2488:
2476:
1873:still be present in lyceum students.
1138:{\displaystyle \varepsilon \in (0,2)}
977:sum to infinity of a geometric series
1999:(18) the result can be either 0 or 1
1835:Hilbert's paradox of the Grand Hotel
3184:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
2365:1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ...
2352:
1849:
13:
2774:Educational Studies in Mathematics
2752:Undergraduate Texts in Mathematics
2391:of two copies of Grandi's series.
1524:
1438:
1337:
1171:
1055:
1000:
900:, but before the advent of modern
898:introduction of calculus in Europe
128:of this series are both 1/2.
56:
14:
3520:
3302:Generalized hypergeometric series
2832:
1960:
1293:{\displaystyle \varepsilon \to 0}
3458:
3457:
3340:
3339:
3312:Lauricella hypergeometric series
3030:
1584:is thus seen to be the value at
912:Relation to the geometric series
3322:Riemann's differential equation
2747:A First Course in Real Analysis
2629:. Scientific American Library.
2576:Bagni, Giorgio T. (June 2005).
821:), subtracting the result from
2482:
2359:Occurrences of Grandi's series
2341:
2158:
2146:
1965:In another study conducted in
1662:
1539:
1529:
1496:
1480:
1451:
1435:
1379:
1363:
1334:
1318:
1284:
1231:
1216:
1192:
1176:
1132:
1120:
997:
960:
945:
802:
790:
616:
586:
498:Treating Grandi's series as a
421:
406:
400:
385:
379:
364:
290:
278:
272:
260:
254:
242:
236:
224:
71:
61:
1:
3317:Modular hypergeometric series
3158:1/4 + 1/16 + 1/64 + 1/256 + ⋯
2568:
2002:(5) the result does not exist
1673:
2599:Davis, Harry F. (May 1989).
2401:1 − 1 + 2 − 6 + 24 − 120 + ⋯
2348:Summation of Grandi's series
2054:
1896:no shock at being told that
1844:
210:would be to treat it like a
7:
3327:Theta hypergeometric series
2822:(4th, reprinted ed.).
2819:A Course of Modern Analysis
2394:
10:
3525:
3209:Infinite arithmetic series
3153:1/2 + 1/4 + 1/8 + 1/16 + ⋯
3148:1/2 − 1/4 + 1/8 − 1/16 + ⋯
2824:Cambridge University Press
2713:Cambridge University Press
2356:
2345:
2026:(2) the result is infinite
1942:to be a plausible result;
1922:. Sierpińska remarks that
1669:History of Grandi's series
1666:
875:divergent geometric series
500:divergent geometric series
3453:
3437:
3421:
3393:
3335:
3292:
3236:
3171:
3140:
3133:
3103:
3072:
3065:
3039:
3028:
2951:
2895:
2886:
2754:. Springer. p. 249.
2505:Protter & Morrey 1991
1300:of series evaluations is
2446:
487:, are sometimes used in
3040:Properties of sequences
2733:Learn from the Masters!
2196:of a different series,
814:{\textstyle -S=(1-S)-1}
485:Eilenberg–Mazur swindle
481:conditional convergence
3504:Mathematical paradoxes
2903:Arithmetic progression
2705:Hobson, E. W. (1907).
2314:
2165:
1898:1 − 1 + 1 − 1 + ··· =
1878:1 − 1 + 1 − 1 + ··· =
1824:
1641:
1625:
1552:
1528:
1479:
1408:
1362:
1294:
1265:
1175:
1139:
1098:
1059:
1025:
967:
966:{\displaystyle (-1,1)}
930:
864:
835:
815:
768:
745:
709:
470:
339:
201:
84:
60:
3294:Hypergeometric series
2908:Geometric progression
2315:
2166:
1825:
1703:absolutely convergent
1680:limit of the sequence
1642:
1605:
1593:analytic continuation
1553:
1508:
1459:
1409:
1342:
1295:
1266:
1155:
1140:
1099:
1039:
1005:
979:can be evaluated via
968:
931:
865:
836:
816:
769:
746:
744:{\displaystyle S=1/2}
710:
471:
340:
202:
85:
40:
3509:Parity (mathematics)
3274:Trigonometric series
3066:Properties of series
2913:Harmonic progression
2646:Mathematics Magazine
2202:
2091:
1996:(26) the result is 0
1712:
1601:
1424:
1307:
1278:
1152:
1111:
986:
942:
920:
863:{\displaystyle 2S=1}
845:
825:
778:
755:
721:
508:
355:
221:
143:
93:is sometimes called
37:
3254:Formal power series
2431:Ramanujan summation
2023:(3) the result is 1
1985:1 − 1 + 1 − 1 + ...
1692:accumulation points
883:1 − 1 + 1 − 1 + ...
132:Nonrigorous methods
126:Ramanujan summation
3429:Luigi Guido Grandi
3052:Monotonic function
2971:Fibonacci sequence
2786:10.1007/BF00240986
2523:, pp. 371–378
2310:
2308:
2161:
2005:(4) the result is
1820:
1637:
1636:
1548:
1458:
1442:
1404:
1341:
1325:
1290:
1261:
1135:
1094:
1004:
963:
926:
860:
831:
811:
764:
741:
705:
703:
466:
335:
212:telescoping series
197:
80:
3471:
3470:
3353:
3352:
3284:Generating series
3232:
3231:
3204:1 − 2 + 4 − 8 + ⋯
3199:1 + 2 + 4 + 8 + ⋯
3194:1 − 2 + 3 − 4 + ⋯
3189:1 + 2 + 3 + 4 + ⋯
3179:1 + 1 + 1 + 1 + ⋯
3129:
3128:
3057:Periodic sequence
3026:
3025:
3011:Triangular number
3001:Pentagonal number
2981:Heptagonal number
2966:Complete sequence
2888:Integer sequences
2841:, Grandi's series
2761:978-0-387-97437-8
2722:978-1-4181-8651-7
2697:978-0-486-66165-0
2636:978-0-7167-6022-1
2610:978-0-486-65973-2
2426:1 − 2 + 4 − 8 + ⋯
2421:1 + 2 + 4 + 8 + ⋯
2416:1 + 2 + 3 + 4 + ⋯
2411:1 − 2 + 3 − 4 + ⋯
2406:1 + 1 + 1 + 1 + ⋯
2322:, giving the sum
2307:
2173:, giving the sum
1976:Liceo Scientifico
1918:1 + 2 + 4 + 8 + ⋯
1443:
1427:
1399:
1326:
1310:
1274:and so the limit
1256:
1235:
1145:, one thus finds
1089:
989:
929:{\displaystyle r}
834:{\displaystyle S}
558:
118:summation methods
28:1 − 1 + 1 − 1 + ⋯
3516:
3494:Geometric series
3489:Divergent series
3461:
3460:
3380:
3373:
3366:
3357:
3356:
3343:
3342:
3269:Dirichlet series
3138:
3137:
3070:
3069:
3034:
3006:Polygonal number
2986:Hexagonal number
2959:
2893:
2892:
2869:
2862:
2855:
2846:
2845:
2827:
2810:Whittaker, E. T.
2805:
2770:Sierpińska, Anna
2765:
2740:
2738:
2726:
2701:
2679:
2661:
2640:
2628:
2614:
2595:
2593:
2587:. Archived from
2582:
2563:
2557:
2548:
2542:
2536:
2530:
2524:
2518:
2507:
2502:
2496:
2486:
2480:
2474:
2465:
2459:
2436:Cesàro summation
2386:
2384:
2383:
2380:
2377:
2366:
2353:Related problems
2337:
2335:
2334:
2331:
2328:
2321:
2319:
2317:
2316:
2311:
2309:
2306:
2305:
2304:
2282:
2271:
2259:
2258:
2246:
2245:
2233:
2232:
2220:
2219:
2195:
2188:
2186:
2185:
2182:
2179:
2172:
2170:
2168:
2167:
2162:
2145:
2128:
2127:
2115:
2114:
2084:geometric series
2081:
2049:
2047:
2046:
2043:
2040:
2020:
2018:
2017:
2014:
2011:
1986:
1956:
1941:
1939:
1938:
1935:
1932:
1921:
1914:
1913:
1911:
1910:
1907:
1904:
1894:
1893:
1891:
1890:
1887:
1884:
1850:Cognitive impact
1829:
1827:
1826:
1821:
1689:
1688:1, 0, 1, 0, ...,
1658:
1656:
1648:
1646:
1644:
1643:
1638:
1635:
1634:
1624:
1619:
1590:
1583:
1581:
1580:
1577:
1574:
1566:complex analysis
1564:In the terms of
1557:
1555:
1554:
1549:
1547:
1546:
1527:
1522:
1504:
1503:
1478:
1473:
1457:
1441:
1413:
1411:
1410:
1405:
1400:
1392:
1387:
1386:
1361:
1356:
1340:
1324:
1299:
1297:
1296:
1291:
1270:
1268:
1267:
1262:
1257:
1255:
1241:
1236:
1234:
1205:
1200:
1199:
1174:
1169:
1144:
1142:
1141:
1136:
1103:
1101:
1100:
1095:
1090:
1088:
1074:
1069:
1068:
1058:
1053:
1035:
1034:
1024:
1019:
1003:
974:
972:
970:
969:
964:
936:in the interval
935:
933:
932:
927:
888:... but its sum
884:
869:
867:
866:
861:
840:
838:
837:
832:
820:
818:
817:
812:
773:
771:
770:
765:
750:
748:
747:
742:
737:
714:
712:
711:
706:
704:
559:
556:
475:
473:
472:
467:
344:
342:
341:
336:
206:
204:
203:
198:
122:Cesàro summation
111:divergent series
97:, after Italian
89:
87:
86:
81:
79:
78:
59:
54:
29:
3524:
3523:
3519:
3518:
3517:
3515:
3514:
3513:
3484:Grandi's series
3474:
3473:
3472:
3467:
3449:
3433:
3417:
3389:
3387:Grandi's series
3384:
3354:
3349:
3331:
3288:
3237:Kinds of series
3228:
3167:
3134:Explicit series
3125:
3099:
3061:
3047:Cauchy sequence
3035:
3022:
2976:Figurate number
2953:
2947:
2938:Powers of three
2882:
2873:
2835:
2830:
2762:
2736:
2723:
2715:. section 331.
2698:
2668:10.2307/2690371
2659:10.1.1.639.6923
2637:
2611:
2591:
2580:
2571:
2566:
2558:
2551:
2543:
2539:
2531:
2527:
2521:Sierpińska 1987
2519:
2510:
2503:
2499:
2491:, p. 307;
2487:
2483:
2475:
2468:
2460:
2453:
2449:
2397:
2381:
2378:
2375:
2374:
2372:
2364:
2361:
2355:
2350:
2344:
2332:
2329:
2326:
2325:
2323:
2300:
2296:
2283:
2272:
2269:
2254:
2250:
2241:
2237:
2228:
2224:
2215:
2211:
2203:
2200:
2199:
2197:
2190:
2183:
2180:
2177:
2176:
2174:
2141:
2123:
2119:
2110:
2106:
2092:
2089:
2088:
2086:
2076:
2057:
2044:
2041:
2038:
2037:
2035:
2015:
2012:
2009:
2008:
2006:
1984:
1963:
1951:
1936:
1933:
1930:
1929:
1927:
1916:
1908:
1905:
1902:
1901:
1899:
1897:
1888:
1885:
1882:
1881:
1879:
1877:
1867:epistemological
1856:Anna Sierpińska
1852:
1847:
1713:
1710:
1709:
1687:
1676:
1671:
1665:
1652:
1650:
1630:
1626:
1620:
1609:
1602:
1599:
1598:
1596:
1585:
1578:
1575:
1572:
1571:
1569:
1542:
1538:
1523:
1512:
1499:
1495:
1474:
1463:
1447:
1431:
1425:
1422:
1421:
1391:
1382:
1378:
1357:
1346:
1330:
1314:
1308:
1305:
1304:
1279:
1276:
1275:
1245:
1240:
1209:
1204:
1195:
1191:
1170:
1159:
1153:
1150:
1149:
1112:
1109:
1108:
1078:
1073:
1064:
1060:
1054:
1043:
1030:
1026:
1020:
1009:
993:
987:
984:
983:
943:
940:
939:
937:
921:
918:
917:
916:For any number
914:
882:
846:
843:
842:
826:
823:
822:
779:
776:
775:
767:{\textstyle -S}
756:
753:
752:
733:
722:
719:
718:
702:
701:
685:
679:
678:
668:
656:
655:
573:
561:
560:
555:
518:
511:
509:
506:
505:
356:
353:
352:
222:
219:
218:
144:
141:
140:
134:
95:Grandi's series
74:
70:
55:
44:
38:
35:
34:
30:, also written
27:
25:infinite series
17:
12:
11:
5:
3522:
3512:
3511:
3506:
3501:
3496:
3491:
3486:
3469:
3468:
3466:
3465:
3454:
3451:
3450:
3448:
3447:
3445:Thomson's lamp
3441:
3439:
3435:
3434:
3432:
3431:
3425:
3423:
3419:
3418:
3416:
3415:
3410:
3405:
3400:
3394:
3391:
3390:
3383:
3382:
3375:
3368:
3360:
3351:
3350:
3348:
3347:
3336:
3333:
3332:
3330:
3329:
3324:
3319:
3314:
3309:
3304:
3298:
3296:
3290:
3289:
3287:
3286:
3281:
3279:Fourier series
3276:
3271:
3266:
3264:Puiseux series
3261:
3259:Laurent series
3256:
3251:
3246:
3240:
3238:
3234:
3233:
3230:
3229:
3227:
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3181:
3175:
3173:
3169:
3168:
3166:
3165:
3160:
3155:
3150:
3144:
3142:
3135:
3131:
3130:
3127:
3126:
3124:
3123:
3118:
3113:
3107:
3105:
3101:
3100:
3098:
3097:
3092:
3087:
3082:
3076:
3074:
3067:
3063:
3062:
3060:
3059:
3054:
3049:
3043:
3041:
3037:
3036:
3029:
3027:
3024:
3023:
3021:
3020:
3019:
3018:
3008:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2962:
2960:
2949:
2948:
2946:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2899:
2897:
2890:
2884:
2883:
2872:
2871:
2864:
2857:
2849:
2843:
2842:
2834:
2833:External links
2831:
2829:
2828:
2806:
2780:(4): 371–396.
2766:
2760:
2741:
2727:
2721:
2702:
2696:
2680:
2652:(5): 307–314.
2641:
2635:
2615:
2609:
2596:
2594:on 2006-12-29.
2572:
2570:
2567:
2565:
2564:
2549:
2537:
2535:, pp. 6–8
2525:
2508:
2497:
2481:
2466:
2450:
2448:
2445:
2444:
2443:
2441:Thomson's lamp
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2396:
2393:
2389:Cauchy product
2357:Main article:
2354:
2351:
2346:Main article:
2343:
2340:
2303:
2299:
2295:
2292:
2289:
2286:
2281:
2278:
2275:
2268:
2265:
2262:
2257:
2253:
2249:
2244:
2240:
2236:
2231:
2227:
2223:
2218:
2214:
2210:
2207:
2160:
2157:
2154:
2151:
2148:
2144:
2140:
2137:
2134:
2131:
2126:
2122:
2118:
2113:
2109:
2105:
2102:
2099:
2096:
2073:
2072:
2065:
2064:
2056:
2053:
2031:
2030:
2029:(30) no answer
2027:
2024:
2021:
2003:
2000:
1997:
1989:
1988:
1962:
1961:Preconceptions
1959:
1948:
1947:
1851:
1848:
1846:
1843:
1831:
1830:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1675:
1672:
1667:Main article:
1664:
1661:
1633:
1629:
1623:
1618:
1615:
1612:
1608:
1595:of the series
1561:is divergent.
1559:
1558:
1545:
1541:
1537:
1534:
1531:
1526:
1521:
1518:
1515:
1511:
1507:
1502:
1498:
1494:
1491:
1488:
1485:
1482:
1477:
1472:
1469:
1466:
1462:
1456:
1453:
1450:
1446:
1440:
1437:
1434:
1430:
1415:
1414:
1403:
1398:
1395:
1390:
1385:
1381:
1377:
1374:
1371:
1368:
1365:
1360:
1355:
1352:
1349:
1345:
1339:
1336:
1333:
1329:
1323:
1320:
1317:
1313:
1289:
1286:
1283:
1272:
1271:
1260:
1254:
1251:
1248:
1244:
1239:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1208:
1203:
1198:
1194:
1190:
1187:
1184:
1181:
1178:
1173:
1168:
1165:
1162:
1158:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1105:
1104:
1093:
1087:
1084:
1081:
1077:
1072:
1067:
1063:
1057:
1052:
1049:
1046:
1042:
1038:
1033:
1029:
1023:
1018:
1015:
1012:
1008:
1002:
999:
996:
992:
962:
959:
956:
953:
950:
947:
925:
913:
910:
906:mathematicians
894:
893:
886:
859:
856:
853:
850:
841:, and solving
830:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
763:
760:
740:
736:
732:
729:
726:
700:
697:
694:
691:
688:
686:
684:
681:
680:
677:
674:
671:
669:
667:
664:
661:
658:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
574:
572:
569:
566:
563:
562:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
519:
517:
514:
513:
477:
476:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
346:
345:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
208:
207:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
133:
130:
91:
90:
77:
73:
69:
66:
63:
58:
53:
50:
47:
43:
15:
9:
6:
4:
3:
2:
3521:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3481:
3479:
3464:
3456:
3455:
3452:
3446:
3443:
3442:
3440:
3436:
3430:
3427:
3426:
3424:
3420:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3395:
3392:
3388:
3381:
3376:
3374:
3369:
3367:
3362:
3361:
3358:
3346:
3338:
3337:
3334:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3299:
3297:
3295:
3291:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3244:Taylor series
3242:
3241:
3239:
3235:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3176:
3174:
3170:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3143:
3139:
3136:
3132:
3122:
3119:
3117:
3114:
3112:
3109:
3108:
3106:
3102:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3077:
3075:
3071:
3068:
3064:
3058:
3055:
3053:
3050:
3048:
3045:
3044:
3042:
3038:
3033:
3017:
3014:
3013:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2963:
2961:
2957:
2950:
2944:
2941:
2939:
2936:
2934:
2933:Powers of two
2931:
2929:
2926:
2924:
2921:
2919:
2918:Square number
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2900:
2898:
2894:
2891:
2889:
2885:
2881:
2877:
2870:
2865:
2863:
2858:
2856:
2851:
2850:
2847:
2840:
2837:
2836:
2825:
2821:
2820:
2815:
2814:Watson, G. N.
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2757:
2753:
2749:
2748:
2742:
2735:
2734:
2728:
2724:
2718:
2714:
2710:
2709:
2703:
2699:
2693:
2689:
2685:
2684:Knopp, Konrad
2681:
2677:
2673:
2669:
2665:
2660:
2655:
2651:
2647:
2642:
2638:
2632:
2627:
2626:
2620:
2619:Devlin, Keith
2616:
2612:
2606:
2602:
2597:
2590:
2586:
2579:
2574:
2573:
2562:, p. 176
2561:
2556:
2554:
2547:, p. 165
2546:
2541:
2534:
2529:
2522:
2517:
2515:
2513:
2506:
2501:
2495:, p. 457
2494:
2490:
2485:
2479:, p. 152
2478:
2473:
2471:
2463:
2458:
2456:
2451:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2398:
2392:
2390:
2370:
2360:
2349:
2339:
2301:
2297:
2293:
2290:
2287:
2284:
2279:
2276:
2273:
2266:
2263:
2260:
2255:
2251:
2247:
2242:
2238:
2234:
2229:
2225:
2221:
2216:
2212:
2208:
2205:
2193:
2155:
2152:
2149:
2142:
2138:
2135:
2132:
2129:
2124:
2120:
2116:
2111:
2107:
2103:
2100:
2097:
2094:
2085:
2079:
2070:
2069:
2068:
2062:
2061:
2060:
2052:
2028:
2025:
2022:
2004:
2001:
1998:
1995:
1994:
1993:
1982:
1981:
1980:
1978:
1977:
1972:
1968:
1958:
1954:
1945:
1944:
1943:
1925:
1919:
1915:or even that
1874:
1872:
1868:
1864:
1861:
1857:
1854:Around 1987,
1842:
1840:
1836:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1708:
1707:
1706:
1704:
1699:
1697:
1693:
1685:
1681:
1670:
1660:
1657:| < 1
1655:
1631:
1627:
1621:
1616:
1613:
1610:
1606:
1594:
1588:
1567:
1562:
1543:
1535:
1532:
1519:
1516:
1513:
1509:
1505:
1500:
1492:
1489:
1486:
1483:
1475:
1470:
1467:
1464:
1460:
1454:
1448:
1432:
1420:
1419:
1418:
1401:
1396:
1393:
1388:
1383:
1375:
1372:
1369:
1366:
1358:
1353:
1350:
1347:
1343:
1331:
1321:
1315:
1303:
1302:
1301:
1287:
1281:
1258:
1252:
1249:
1246:
1242:
1237:
1228:
1225:
1222:
1219:
1213:
1210:
1206:
1201:
1196:
1188:
1185:
1182:
1179:
1166:
1163:
1160:
1156:
1148:
1147:
1146:
1129:
1126:
1123:
1117:
1114:
1091:
1085:
1082:
1079:
1075:
1070:
1065:
1061:
1050:
1047:
1044:
1040:
1036:
1031:
1027:
1021:
1016:
1013:
1010:
1006:
994:
982:
981:
980:
978:
957:
954:
951:
948:
923:
909:
907:
903:
899:
891:
887:
880:
879:
878:
876:
871:
857:
854:
851:
848:
828:
808:
805:
799:
796:
793:
787:
784:
781:
761:
758:
738:
734:
730:
727:
724:
717:resulting in
715:
698:
695:
692:
689:
687:
682:
675:
672:
670:
665:
662:
659:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
613:
610:
607:
604:
601:
598:
595:
592:
589:
583:
580:
577:
575:
570:
567:
564:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
520:
515:
503:
501:
496:
494:
490:
486:
482:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
418:
415:
412:
409:
403:
397:
394:
391:
388:
382:
376:
373:
370:
367:
361:
358:
351:
350:
349:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
287:
284:
281:
275:
269:
266:
263:
257:
251:
248:
245:
239:
233:
230:
227:
217:
216:
215:
213:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
139:
138:
137:
129:
127:
123:
119:
114:
112:
108:
105:, and priest
104:
100:
99:mathematician
96:
75:
67:
64:
51:
48:
45:
41:
33:
32:
31:
26:
22:
3386:
3249:Power series
3183:
2991:Lucas number
2943:Powers of 10
2923:Cubic number
2817:
2777:
2773:
2746:
2732:
2707:
2687:
2649:
2645:
2624:
2600:
2589:the original
2584:
2560:Lehmann 1995
2545:Lehmann 1995
2540:
2528:
2500:
2484:
2464:, p. 77
2362:
2191:
2077:
2074:
2066:
2058:
2032:
1990:
1974:
1964:
1949:
1923:
1875:
1870:
1853:
1832:
1700:
1684:partial sums
1677:
1653:
1586:
1563:
1560:
1416:
1273:
1106:
915:
895:
889:
872:
716:
504:
497:
478:
347:
209:
135:
115:
107:Guido Grandi
94:
92:
18:
3413:Occurrences
3116:Conditional
3104:Convergence
3095:Telescoping
3080:Alternating
2996:Pell number
2462:Devlin 1994
2363:The series
2342:Summability
1839:permutation
1663:Early ideas
885:has no sum.
881:The series
489:knot theory
103:philosopher
21:mathematics
3499:1 (number)
3478:Categories
3141:Convergent
3085:Convergent
2569:References
2533:Bagni 2005
2493:Knopp 1990
2489:Kline 1983
2477:Davis 1989
1674:Divergence
3408:Summation
3403:Education
3172:Divergent
3090:Divergent
2952:Advanced
2928:Factorial
2876:Sequences
2802:144880659
2690:. Dover.
2686:(1990) .
2654:CiteSeerX
2603:. Dover.
2264:⋯
2261:−
2235:−
2209:−
2133:⋯
2117:−
2098:−
2055:Prospects
1845:Education
1818:⋯
1815:−
1797:−
1791:−
1773:−
1767:−
1749:−
1743:−
1696:divergent
1607:∑
1533:−
1525:∞
1510:∑
1493:ε
1484:−
1461:∑
1452:→
1449:ε
1439:∞
1436:→
1376:ε
1367:−
1344:∑
1338:∞
1335:→
1319:→
1316:ε
1285:→
1282:ε
1253:ε
1250:−
1229:ε
1220:−
1214:−
1189:ε
1180:−
1172:∞
1157:∑
1118:∈
1115:ε
1083:−
1056:∞
1041:∑
1007:∑
1001:∞
998:→
949:−
806:−
797:−
782:−
759:−
663:−
647:…
638:−
626:−
614:…
605:−
593:−
584:−
568:−
550:…
541:−
529:−
458:…
428:…
410:−
389:−
368:−
327:…
297:…
285:−
267:−
249:−
231:−
195:…
186:−
174:−
162:−
150:−
65:−
57:∞
42:∑
3463:Category
3345:Category
3111:Absolute
2826:. § 2.1.
2816:(1962).
2621:(1994).
2395:See also
1953:0.999...
1924:a priori
1107:For any
557: so
124:and the
3438:Related
3398:History
3121:Uniform
2794:3482354
2676:2690371
2385:
2373:
2336:
2324:
2320:
2198:
2187:
2175:
2171:
2087:
2082:of the
2048:
2036:
2019:
2007:
1967:Treviso
1940:
1928:
1912:
1900:
1892:
1880:
1837:) is a
1682:of its
1647:
1597:
1591:of the
1582:
1570:
973:
938:
892:be 1/2.
774:(from (
493:algebra
3422:People
3073:Series
2880:series
2800:
2792:
2758:
2719:
2694:
2674:
2656:
2633:
2607:
1863:lyceum
1860:Warsaw
1651:|
975:, the
902:rigour
890:should
23:, the
3016:array
2896:Basic
2798:S2CID
2790:JSTOR
2737:(PDF)
2672:JSTOR
2592:(PDF)
2581:(PDF)
2447:Notes
2369:up to
1971:Italy
2956:list
2878:and
2756:ISBN
2717:ISBN
2692:ISBN
2631:ISBN
2605:ISBN
1920:= −1
1589:= −1
491:and
2782:doi
2664:doi
2194:= 1
2080:= 1
1955:= 1
1871:may
1445:lim
1429:lim
1328:lim
1312:lim
991:lim
19:In
3480::
2812:;
2796:.
2788:.
2778:18
2776:.
2750:.
2711:.
2670:.
2662:.
2650:56
2648:.
2583:.
2552:^
2511:^
2469:^
2454:^
1969:,
1698:.
1659:.
1568:,
908:.
870:.
464:1.
333:0.
101:,
3379:e
3372:t
3365:v
2958:)
2954:(
2868:e
2861:t
2854:v
2804:.
2784::
2764:.
2725:.
2700:.
2678:.
2666::
2639:.
2613:.
2382:4
2379:/
2376:1
2367:(
2333:3
2330:/
2327:2
2302:2
2298:x
2294:+
2291:x
2288:+
2285:1
2280:x
2277:+
2274:1
2267:=
2256:6
2252:x
2248:+
2243:5
2239:x
2230:3
2226:x
2222:+
2217:2
2213:x
2206:1
2192:x
2184:2
2181:/
2178:1
2159:)
2156:x
2153:+
2150:1
2147:(
2143:/
2139:1
2136:=
2130:+
2125:3
2121:x
2112:2
2108:x
2104:+
2101:x
2095:1
2078:x
2045:2
2042:/
2039:1
2016:2
2013:/
2010:1
1937:2
1934:/
1931:1
1909:2
1906:/
1903:1
1889:2
1886:/
1883:1
1812:1
1809:+
1806:1
1803:+
1800:1
1794:1
1788:1
1785:+
1782:1
1779:+
1776:1
1770:1
1764:1
1761:+
1758:1
1755:+
1752:1
1746:1
1740:1
1737:+
1734:1
1731:+
1728:1
1725:+
1722:1
1719:+
1716:1
1654:z
1632:n
1628:z
1622:N
1617:0
1614:=
1611:n
1587:z
1579:2
1576:/
1573:1
1544:n
1540:)
1536:1
1530:(
1520:0
1517:=
1514:n
1506:=
1501:n
1497:)
1490:+
1487:1
1481:(
1476:N
1471:0
1468:=
1465:n
1455:0
1433:N
1402:.
1397:2
1394:1
1389:=
1384:n
1380:)
1373:+
1370:1
1364:(
1359:N
1354:0
1351:=
1348:n
1332:N
1322:0
1288:0
1259:,
1247:2
1243:1
1238:=
1232:)
1226:+
1223:1
1217:(
1211:1
1207:1
1202:=
1197:n
1193:)
1186:+
1183:1
1177:(
1167:0
1164:=
1161:n
1133:)
1130:2
1127:,
1124:0
1121:(
1092:.
1086:r
1080:1
1076:1
1071:=
1066:n
1062:r
1051:0
1048:=
1045:n
1037:=
1032:n
1028:r
1022:N
1017:0
1014:=
1011:n
995:N
961:)
958:1
955:,
952:1
946:(
924:r
858:1
855:=
852:S
849:2
829:S
809:1
803:)
800:S
794:1
791:(
788:=
785:S
762:S
739:2
735:/
731:1
728:=
725:S
699:,
696:S
693:2
690:=
683:1
676:S
673:=
666:S
660:1
653:S
650:=
644:+
641:1
635:1
632:+
629:1
623:1
620:=
617:)
611:+
608:1
602:1
599:+
596:1
590:1
587:(
581:1
578:=
571:S
565:1
553:,
547:+
544:1
538:1
535:+
532:1
526:1
523:=
516:S
461:=
455:+
452:0
449:+
446:0
443:+
440:0
437:+
434:1
431:=
425:+
422:)
419:1
416:+
413:1
407:(
404:+
401:)
398:1
395:+
392:1
386:(
383:+
380:)
377:1
374:+
371:1
365:(
362:+
359:1
330:=
324:+
321:0
318:+
315:0
312:+
309:0
306:+
303:0
300:=
294:+
291:)
288:1
282:1
279:(
276:+
273:)
270:1
264:1
261:(
258:+
255:)
252:1
246:1
243:(
240:+
237:)
234:1
228:1
225:(
192:+
189:1
183:1
180:+
177:1
171:1
168:+
165:1
159:1
156:+
153:1
147:1
76:n
72:)
68:1
62:(
52:0
49:=
46:n
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