619:
3191:
31:
3500:
696:
684:
2052:
of the first and last individual terms. This correspondence follows the usual pattern that any arithmetic sequence is a sequence of logarithms of terms of a geometric sequence and any geometric sequence is a sequence of exponentiations of terms of an arithmetic sequence. Sums of logarithms correspond
557:
When the common ratio of a geometric sequence is positive, the sequence's terms will all share the sign of the first term. When the common ratio of a geometric sequence is negative, the sequence's terms alternate between positive and negative; this is called an alternating sequence. For instance the
2040:
1051:
2712:
1422:
2205:
1294:
1792:
1870:
186:
558:
sequence 1, −3, 9, −27, 81, −243, ... is an alternating geometric sequence with an initial value of 1 and a common ratio of −3. When the initial term and common ratio are complex numbers, the terms'
2310:
2399:
2479:
1168:
610:: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression yields an arithmetic progression.
2556:
2519:
524:
735:
in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric sequence forms a geometric series. Each term is therefore the
69:. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2.
925:
355:
1492:
289:
414:
976:
1528:
2611:
1572:
550:
440:
2582:
687:
The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square (1/2×
2086:
1864:
1082:
2106:
1834:
1814:
1678:
1622:
1602:
965:
945:
3382:
1660:
The infinite product of a geometric progression is the product of all of its terms. The partial product of a geometric progression up to the term with power
585:. If the absolute value of the common ratio equals 1, the terms will stay the same size indefinitely, though their signs or complex arguments may change.
2618:
2975:
2966:
1685:
2958:
3025:
3372:
1304:
2954:
2729:(c. 2900 – c. 2350 BC), identified as MS 3047, contains a geometric progression with base 3 and multiplier 1/2. It has been suggested to be
2114:
3465:
1195:
446:
2738:
968:
719:
is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.
3306:
1840:
of the partial progression's first and last individual terms and then raising that mean to the power given by the number of terms
598:
93:
3316:
2871:
2217:
2726:
2326:
3480:
3311:
3071:
3018:
2790:
17:
2414:
1090:
691:= 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
3460:
2924:
2035:{\displaystyle \prod _{k=0}^{n}ar^{k}=a^{n+1}r^{n(n+1)/2}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0.}
588:
Geometric progressions show exponential growth or exponential decline, as opposed to arithmetic progressions showing
3470:
811:
Geometric series have been applied to model a wide variety of natural phenomena and social phenomena, such as the
3362:
3352:
2858:. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153.
577:. If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approach
2528:
2491:
2773:
457:
3524:
3475:
3377:
3011:
2947:
2796:
857:
3529:
3503:
2937:
300:
2854:
Friberg, Jöran (2007). "MS 3047: An Old
Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.).
3485:
2942:
2963:
1429:
3367:
2779:
589:
241:
369:
65:
where each term after the first is found by multiplying the previous one by a fixed number called the
42:
of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
3357:
3347:
3337:
812:
573:
of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via an
1170:
are called "finite geometric series" in certain branches of mathematics, especially in 19th century
1046:{\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }
2811: – Standard guidelines for choosing exact product dimensions within a given set of constraints
2979:
819:, the decay of radioactive carbon-14 atoms where the common ratio between terms is defined by the
796:. Geometric series have further served as prototypes in the study of mathematical objects such as
3452:
3274:
3114:
3061:
2820:
2767:
1629:
1186:
563:
1497:
1056:
is geometric because each successive term can be obtained by multiplying the previous term by
788:
and thus came to be crucial references for investigations of convergence, for instance in the
38:) up to 6 iterations deep. The first block is a unit block and the dashed line represents the
3321:
2814:
2587:
1537:
593:
529:
419:
2561:
3432:
3269:
3038:
2881:
2802:
2784:
2749:
2064:
1625:
756:
732:
603:
1843:
8:
3412:
3279:
2045:
1640:
1531:
1059:
805:
801:
793:
622:
618:
361:
3342:
3253:
3238:
3210:
3190:
3129:
2091:
1819:
1799:
1663:
1644:
1607:
1587:
950:
930:
820:
582:
3442:
3243:
3215:
3169:
3159:
3139:
3124:
2989:
2986:
2920:
2900:
2867:
2737:. It is the only known record of a geometric progression from before the time of old
836:
781:
740:
574:
3427:
3248:
3174:
3164:
3144:
3046:
2859:
2808:
1637:
785:
728:
676:
559:
204:
3205:
3134:
2970:
2877:
2793: – Progression formed by taking the reciprocals of an arithmetic progression
2049:
816:
744:
3437:
3422:
3417:
3096:
3081:
2707:{\displaystyle P_{n}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0}
1837:
1581:
832:
828:
736:
570:
73:
2863:
3518:
3402:
3076:
2755:
2320:
is the sum of an arithmetic sequence. Substituting the formula for that sum,
1633:
797:
750:
Geometric series have been studied in mathematics from at least the time of
715:) = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the
3407:
3149:
3091:
2955:
Derivation of formulas for sum of finite and infinite geometric progression
765:
84:
3154:
3101:
2895:
1577:
1175:
824:
724:
716:
55:
554:
This is a second order nonlinear recurrence with constant coefficients.
3003:
1636:
geometric series, and, most generally, geometric series of elements of
1417:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}
1179:
840:
789:
761:
2200:{\displaystyle P_{n}=a\cdot ar\cdot ar^{2}\cdots ar^{n-1}\cdot ar^{n}}
707:= 1/9) shown as areas of purple squares. The total purple area is S =
695:
34:
Diagram illustrating three basic geometric sequences of the pattern 1(
30:
3086:
2994:
2758:, see the article for details) and give several of their properties.
2734:
2048:: the sum of an arithmetic sequence is the number of terms times the
1576:
Though geometric series are most commonly found and applied with the
1289:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots =\sum _{k=0}^{\infty }ar^{k}}
844:
607:
2044:
This corresponds to a similar property of sums of terms of a finite
1299:
and the corresponding expression for the finite geometric series is
3034:
1648:
1171:
848:
777:
773:
683:
578:
452:
Geometric sequences also satisfy the nonlinear recurrence relation
58:
2745:
843:
where the common ratio could be determined by a combination of
751:
62:
739:
of its two neighbouring terms, similar to how the terms in an
2730:
1787:{\displaystyle \prod _{k=0}^{n}ar^{(k)}=a^{n+1}r^{n(n+1)/2}.}
769:
768:, particularly in calculating areas and volumes of geometric
1836:
are positive real numbers, this is equivalent to taking the
831:
where the common ratio could be determined by the odds of a
2776: – Mathematical sequence satisfying a specific pattern
2211:
Carrying out the multiplications and gathering like terms,
181:{\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }
2984:
967:
is the common ratio between adjacent terms. For example,
625:
of the formula for the sum of a geometric series –
2856:
A remarkable collection of
Babylonian mathematical texts
2844:
Belmont, California, Wadsworth
Publishing, p. 566, 1970.
1624:, there are also important results and applications for
602:. The two kinds of progression are related through the
592:
growth or linear decline. This comparison was taken by
202:
The sum of a geometric progression's terms is called a
2834:
2532:
2495:
2305:{\displaystyle P_{n}=a^{n+1}r^{1+2+3+\cdots +(n-1)+n}}
3383:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
3373:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
2717:
which is the formula in terms of the geometric mean.
2621:
2590:
2564:
2531:
2494:
2417:
2329:
2220:
2117:
2094:
2067:
1873:
1846:
1822:
1802:
1688:
1666:
1610:
1590:
1540:
1500:
1432:
1307:
1198:
1093:
1062:
979:
953:
933:
860:
532:
460:
422:
372:
303:
244:
96:
2799: – Divergent sum of all positive unit fractions
2787: – Mathematical function, denoted exp(x) or e^x
815:where the common ratio between terms is defined by
220:th term of a geometric sequence with initial value
87:and 3. The general form of a geometric sequence is
2906:(2nd ed. ed.). New York: Dover Publications.
2899:
2706:
2605:
2576:
2550:
2513:
2473:
2394:{\displaystyle P_{n}=a^{n+1}r^{\frac {n(n+1)}{2}}}
2393:
2304:
2199:
2100:
2080:
2034:
1858:
1828:
1808:
1786:
1672:
1616:
1596:
1566:
1522:
1486:
1416:
1288:
1162:
1076:
1045:
959:
939:
919:
544:
518:
434:
408:
349:
283:
180:
1189:expression for the infinite geometric series is
3516:
2474:{\displaystyle P_{n}=(ar^{\frac {n}{2}})^{n+1}.}
1163:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}}
2817: – British political economist (1766–1834)
764:further advanced the study through his work on
2842:Calculus and Analytic Geometry, Second Edition
3019:
854:In general, a geometric series is written as
3466:Hypergeometric function of a matrix argument
2754:analyze geometric progressions (such as the
772:(for instance calculating the area inside a
447:linear recurrence with constant coefficients
3322:1 + 1/2 + 1/3 + ... (Riemann zeta function)
3026:
3012:
2770: – Sequence of equally spaced numbers
2551:{\displaystyle \textstyle {\sqrt {r^{2}}}}
2514:{\displaystyle \textstyle {\sqrt {a^{2}}}}
792:for convergence and in the definitions of
780:, they were paradigmatic examples of both
3378:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
2976:Nice Proof of a Geometric Progression Sum
1039:
1035:
1024:
1020:
1009:
1005:
994:
990:
519:{\displaystyle a_{n}=a_{n-1}^{2}/a_{n-2}}
389:
327:
261:
3033:
1426:Any finite geometric series has the sum
760:, which explored geometric proportions.
694:
682:
617:
29:
2902:The Thirteen Books of Euclid's Elements
2853:
599:An Essay on the Principle of Population
360:Geometric sequences satisfy the linear
14:
3517:
920:{\displaystyle a+ar+ar^{2}+ar^{3}+...}
776:). In the early development of modern
699:Another geometric series (coefficient
596:as the mathematical foundation of his
3007:
2985:
2894:
2408:One can rearrange this expression to
2053:to products of exponentiated values.
1530:the infinite series converges to the
350:{\displaystyle a_{n}=a_{m}\,r^{n-m}.}
72:Examples of a geometric sequence are
2727:Early Dynastic Period in Mesopotamia
3343:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
613:
445:This is a first order, homogeneous
24:
2088:represent the product up to power
1487:{\displaystyle a(1-r^{n+1})/(1-r)}
1268:
747:of their two neighbouring terms.
25:
3541:
3461:Generalized hypergeometric series
2930:
284:{\displaystyle a_{n}=a\,r^{n-1},}
3499:
3498:
3471:Lauricella hypergeometric series
3189:
2964:Geometric Progression Calculator
2823: – Probability distribution
675:This section is an excerpt from
409:{\displaystyle a_{n}=r\,a_{n-1}}
27:Mathematical sequence of numbers
3481:Riemann's differential equation
2888:
2847:
2774:Arithmetico-geometric sequence
2663:
2635:
2453:
2431:
2381:
2369:
2291:
2279:
1991:
1963:
1947:
1935:
1768:
1756:
1724:
1718:
1561:
1549:
1510:
1502:
1481:
1469:
1461:
1436:
13:
1:
3476:Modular hypergeometric series
3317:1/4 + 1/16 + 1/64 + 1/256 + ⋯
2827:
2558:though this is not valid for
211:
2405:which concludes the proof.
7:
3486:Theta hypergeometric series
2943:Encyclopedia of Mathematics
2761:
1087:Truncated geometric series
79:of a fixed non-zero number
39:
10:
3546:
3368:Infinite arithmetic series
3312:1/2 + 1/4 + 1/8 + 1/16 + ⋯
3307:1/2 − 1/4 + 1/8 − 1/16 + ⋯
2780:Linear difference equation
2720:
1655:
1632:-valued geometric series,
1628:-valued geometric series,
674:
3494:
3451:
3395:
3330:
3299:
3292:
3262:
3231:
3224:
3198:
3187:
3110:
3054:
3045:
2864:10.1007/978-0-387-48977-3
813:expansion of the universe
2056:
1523:{\displaystyle |r|<1}
1182:and their applications.
947:is the initial term and
195:is the common ratio and
3199:Properties of sequences
2938:"Geometric progression"
2725:A clay tablet from the
2606:{\displaystyle r<0,}
2108:. Written out in full,
1567:{\displaystyle a/(1-r)}
703:= 4/9 and common ratio
641:term vanishes, leaving
545:{\displaystyle n>2.}
435:{\displaystyle n>1.}
199:is the initial value.
3062:Arithmetic progression
2821:Geometric distribution
2768:Arithmetic progression
2741:beginning in 2000 BC.
2739:Babylonian mathematics
2708:
2607:
2578:
2577:{\displaystyle a<0}
2552:
2515:
2475:
2395:
2306:
2201:
2102:
2082:
2036:
1894:
1860:
1830:
1810:
1788:
1709:
1674:
1618:
1598:
1568:
1524:
1488:
1418:
1397:
1290:
1272:
1187:capital-sigma notation
1164:
1078:
1047:
961:
941:
921:
821:half-life of carbon-14
720:
692:
671:
564:arithmetic progression
546:
520:
436:
410:
351:
285:
182:
43:
3453:Hypergeometric series
3067:Geometric progression
2815:Thomas Robert Malthus
2744:Books VIII and IX of
2709:
2608:
2579:
2553:
2516:
2476:
2396:
2307:
2202:
2103:
2083:
2081:{\displaystyle P_{n}}
2037:
1874:
1861:
1831:
1811:
1789:
1689:
1675:
1619:
1599:
1569:
1525:
1489:
1419:
1377:
1291:
1252:
1165:
1079:
1048:
962:
942:
922:
806:perturbation theories
698:
686:
621:
547:
521:
437:
411:
352:
286:
183:
48:geometric progression
33:
3525:Sequences and series
3433:Trigonometric series
3225:Properties of series
3072:Harmonic progression
2805: – Infinite sum
2791:Harmonic progression
2785:Exponential function
2619:
2588:
2562:
2529:
2492:
2415:
2327:
2218:
2115:
2092:
2065:
1871:
1859:{\displaystyle n+1.}
1844:
1820:
1800:
1686:
1664:
1608:
1588:
1538:
1498:
1430:
1305:
1196:
1091:
1060:
977:
951:
931:
858:
802:generating functions
794:rates of convergence
604:exponential function
530:
458:
420:
370:
301:
242:
94:
3530:Mathematical series
3413:Formal power series
2915:Hall & Knight,
2840:Riddle, Douglas F.
2733:, from the city of
2046:arithmetic sequence
1077:{\displaystyle 1/2}
623:Proof without words
494:
362:recurrence relation
3211:Monotonic function
3130:Fibonacci sequence
2990:"Geometric Series"
2987:Weisstein, Eric W.
2969:2008-12-27 at the
2704:
2603:
2574:
2548:
2547:
2511:
2510:
2471:
2391:
2302:
2197:
2098:
2078:
2032:
1856:
1826:
1806:
1784:
1670:
1638:abstract algebraic
1614:
1594:
1564:
1520:
1484:
1414:
1286:
1160:
1074:
1043:
957:
937:
917:
721:
693:
672:
631:| < 1 and
583:exponential growth
542:
526:for every integer
516:
474:
432:
416:for every integer
406:
347:
281:
178:
52:geometric sequence
50:, also known as a
44:
18:Geometric sequence
3512:
3511:
3443:Generating series
3391:
3390:
3363:1 − 2 + 4 − 8 + ⋯
3358:1 + 2 + 4 + 8 + ⋯
3353:1 − 2 + 3 − 4 + ⋯
3348:1 + 2 + 3 + 4 + ⋯
3338:1 + 1 + 1 + 1 + ⋯
3288:
3287:
3216:Periodic sequence
3185:
3184:
3170:Triangular number
3160:Pentagonal number
3140:Heptagonal number
3125:Complete sequence
3047:Integer sequences
2873:978-0-387-34543-7
2681:
2660:
2545:
2508:
2449:
2388:
2101:{\displaystyle n}
2009:
1988:
1829:{\displaystyle r}
1809:{\displaystyle a}
1673:{\displaystyle n}
1617:{\displaystyle r}
1597:{\displaystyle a}
1033:
1018:
1003:
988:
960:{\displaystyle r}
940:{\displaystyle a}
817:Hubble's constant
782:convergent series
741:arithmetic series
614:Geometric series
575:exponential decay
560:complex arguments
231:and common ratio
174:
155:
136:
117:
105:
16:(Redirected from
3537:
3502:
3501:
3428:Dirichlet series
3297:
3296:
3229:
3228:
3193:
3165:Polygonal number
3145:Hexagonal number
3118:
3052:
3051:
3028:
3021:
3014:
3005:
3004:
3000:
2999:
2951:
2908:
2907:
2905:
2896:Heath, Thomas L.
2892:
2886:
2885:
2851:
2845:
2838:
2809:Preferred number
2713:
2711:
2710:
2705:
2682:
2679:
2677:
2676:
2661:
2659:
2658:
2649:
2648:
2639:
2631:
2630:
2612:
2610:
2609:
2604:
2583:
2581:
2580:
2575:
2557:
2555:
2554:
2549:
2546:
2544:
2543:
2534:
2524:
2520:
2518:
2517:
2512:
2509:
2507:
2506:
2497:
2487:
2480:
2478:
2477:
2472:
2467:
2466:
2451:
2450:
2442:
2427:
2426:
2400:
2398:
2397:
2392:
2390:
2389:
2384:
2364:
2358:
2357:
2339:
2338:
2319:
2316:The exponent of
2311:
2309:
2308:
2303:
2301:
2300:
2249:
2248:
2230:
2229:
2206:
2204:
2203:
2198:
2196:
2195:
2180:
2179:
2158:
2157:
2127:
2126:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2077:
2076:
2041:
2039:
2038:
2033:
2010:
2007:
2005:
2004:
1989:
1987:
1986:
1977:
1976:
1967:
1959:
1958:
1954:
1926:
1925:
1907:
1906:
1893:
1888:
1865:
1863:
1862:
1857:
1835:
1833:
1832:
1827:
1815:
1813:
1812:
1807:
1793:
1791:
1790:
1785:
1780:
1779:
1775:
1747:
1746:
1728:
1727:
1708:
1703:
1679:
1677:
1676:
1671:
1623:
1621:
1620:
1615:
1603:
1601:
1600:
1595:
1573:
1571:
1570:
1565:
1548:
1529:
1527:
1526:
1521:
1513:
1505:
1493:
1491:
1490:
1485:
1468:
1460:
1459:
1423:
1421:
1420:
1415:
1410:
1409:
1396:
1391:
1373:
1372:
1351:
1350:
1335:
1334:
1295:
1293:
1292:
1287:
1285:
1284:
1271:
1266:
1242:
1241:
1226:
1225:
1169:
1167:
1166:
1161:
1159:
1158:
1137:
1136:
1121:
1120:
1083:
1081:
1080:
1075:
1070:
1052:
1050:
1049:
1044:
1034:
1026:
1019:
1011:
1004:
996:
989:
981:
966:
964:
963:
958:
946:
944:
943:
938:
926:
924:
923:
918:
904:
903:
888:
887:
786:divergent series
745:arithmetic means
729:geometric series
690:
677:Geometric series
670:
669:
667:
666:
660:
657:
636:
635:→ ∞,
551:
549:
548:
543:
525:
523:
522:
517:
515:
514:
499:
493:
488:
470:
469:
441:
439:
438:
433:
415:
413:
412:
407:
405:
404:
382:
381:
356:
354:
353:
348:
343:
342:
326:
325:
313:
312:
290:
288:
287:
282:
277:
276:
254:
253:
205:geometric series
187:
185:
184:
179:
172:
168:
167:
153:
149:
148:
134:
130:
129:
115:
103:
21:
3545:
3544:
3540:
3539:
3538:
3536:
3535:
3534:
3515:
3514:
3513:
3508:
3490:
3447:
3396:Kinds of series
3387:
3326:
3293:Explicit series
3284:
3258:
3220:
3206:Cauchy sequence
3194:
3181:
3135:Figurate number
3112:
3106:
3097:Powers of three
3041:
3032:
2971:Wayback Machine
2936:
2933:
2912:
2911:
2893:
2889:
2874:
2852:
2848:
2839:
2835:
2830:
2803:Infinite series
2797:Harmonic series
2764:
2723:
2680: for
2678:
2666:
2662:
2654:
2650:
2644:
2640:
2638:
2626:
2622:
2620:
2617:
2616:
2589:
2586:
2585:
2563:
2560:
2559:
2539:
2535:
2533:
2530:
2527:
2526:
2522:
2502:
2498:
2496:
2493:
2490:
2489:
2485:
2456:
2452:
2441:
2437:
2422:
2418:
2416:
2413:
2412:
2365:
2363:
2359:
2347:
2343:
2334:
2330:
2328:
2325:
2324:
2317:
2254:
2250:
2238:
2234:
2225:
2221:
2219:
2216:
2215:
2191:
2187:
2169:
2165:
2153:
2149:
2122:
2118:
2116:
2113:
2112:
2093:
2090:
2089:
2072:
2068:
2066:
2063:
2062:
2059:
2050:arithmetic mean
2008: for
2006:
1994:
1990:
1982:
1978:
1972:
1968:
1966:
1950:
1931:
1927:
1915:
1911:
1902:
1898:
1889:
1878:
1872:
1869:
1868:
1845:
1842:
1841:
1821:
1818:
1817:
1801:
1798:
1797:
1771:
1752:
1748:
1736:
1732:
1717:
1713:
1704:
1693:
1687:
1684:
1683:
1665:
1662:
1661:
1658:
1653:
1652:
1609:
1606:
1605:
1589:
1586:
1585:
1582:complex numbers
1544:
1539:
1536:
1535:
1509:
1501:
1499:
1496:
1495:
1464:
1449:
1445:
1431:
1428:
1427:
1405:
1401:
1392:
1381:
1368:
1364:
1346:
1342:
1330:
1326:
1306:
1303:
1302:
1280:
1276:
1267:
1256:
1237:
1233:
1221:
1217:
1197:
1194:
1193:
1154:
1150:
1132:
1128:
1116:
1112:
1092:
1089:
1088:
1066:
1061:
1058:
1057:
1025:
1010:
995:
980:
978:
975:
974:
952:
949:
948:
932:
929:
928:
899:
895:
883:
879:
859:
856:
855:
837:economic values
829:games of chance
688:
680:
661:
658:
653:
652:
650:
648:
642:
626:
616:
531:
528:
527:
504:
500:
495:
489:
478:
465:
461:
459:
456:
455:
421:
418:
417:
394:
390:
377:
373:
371:
368:
367:
332:
328:
321:
317:
308:
304:
302:
299:
298:
294:and in general
266:
262:
249:
245:
243:
240:
239:
230:
214:
163:
159:
144:
140:
125:
121:
95:
92:
91:
28:
23:
22:
15:
12:
11:
5:
3543:
3533:
3532:
3527:
3510:
3509:
3507:
3506:
3495:
3492:
3491:
3489:
3488:
3483:
3478:
3473:
3468:
3463:
3457:
3455:
3449:
3448:
3446:
3445:
3440:
3438:Fourier series
3435:
3430:
3425:
3423:Puiseux series
3420:
3418:Laurent series
3415:
3410:
3405:
3399:
3397:
3393:
3392:
3389:
3388:
3386:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3340:
3334:
3332:
3328:
3327:
3325:
3324:
3319:
3314:
3309:
3303:
3301:
3294:
3290:
3289:
3286:
3285:
3283:
3282:
3277:
3272:
3266:
3264:
3260:
3259:
3257:
3256:
3251:
3246:
3241:
3235:
3233:
3226:
3222:
3221:
3219:
3218:
3213:
3208:
3202:
3200:
3196:
3195:
3188:
3186:
3183:
3182:
3180:
3179:
3178:
3177:
3167:
3162:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3121:
3119:
3108:
3107:
3105:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3058:
3056:
3049:
3043:
3042:
3031:
3030:
3023:
3016:
3008:
3002:
3001:
2982:
2973:
2961:
2952:
2932:
2931:External links
2929:
2928:
2927:
2919:, p. 39,
2917:Higher Algebra
2910:
2909:
2887:
2872:
2846:
2832:
2831:
2829:
2826:
2825:
2824:
2818:
2812:
2806:
2800:
2794:
2788:
2782:
2777:
2771:
2763:
2760:
2722:
2719:
2715:
2714:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2675:
2672:
2669:
2665:
2657:
2653:
2647:
2643:
2637:
2634:
2629:
2625:
2602:
2599:
2596:
2593:
2573:
2570:
2567:
2542:
2538:
2505:
2501:
2482:
2481:
2470:
2465:
2462:
2459:
2455:
2448:
2445:
2440:
2436:
2433:
2430:
2425:
2421:
2403:
2402:
2387:
2383:
2380:
2377:
2374:
2371:
2368:
2362:
2356:
2353:
2350:
2346:
2342:
2337:
2333:
2314:
2313:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2253:
2247:
2244:
2241:
2237:
2233:
2228:
2224:
2209:
2208:
2194:
2190:
2186:
2183:
2178:
2175:
2172:
2168:
2164:
2161:
2156:
2152:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2125:
2121:
2097:
2075:
2071:
2058:
2055:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2003:
2000:
1997:
1993:
1985:
1981:
1975:
1971:
1965:
1962:
1957:
1953:
1949:
1946:
1943:
1940:
1937:
1934:
1930:
1924:
1921:
1918:
1914:
1910:
1905:
1901:
1897:
1892:
1887:
1884:
1881:
1877:
1855:
1852:
1849:
1838:geometric mean
1825:
1805:
1783:
1778:
1774:
1770:
1767:
1764:
1761:
1758:
1755:
1751:
1745:
1742:
1739:
1735:
1731:
1726:
1723:
1720:
1716:
1712:
1707:
1702:
1699:
1696:
1692:
1669:
1657:
1654:
1613:
1593:
1563:
1560:
1557:
1554:
1551:
1547:
1543:
1519:
1516:
1512:
1508:
1504:
1483:
1480:
1477:
1474:
1471:
1467:
1463:
1458:
1455:
1452:
1448:
1444:
1441:
1438:
1435:
1413:
1408:
1404:
1400:
1395:
1390:
1387:
1384:
1380:
1376:
1371:
1367:
1363:
1360:
1357:
1354:
1349:
1345:
1341:
1338:
1333:
1329:
1325:
1322:
1319:
1316:
1313:
1310:
1297:
1296:
1283:
1279:
1275:
1270:
1265:
1262:
1259:
1255:
1251:
1248:
1245:
1240:
1236:
1232:
1229:
1224:
1220:
1216:
1213:
1210:
1207:
1204:
1201:
1157:
1153:
1149:
1146:
1143:
1140:
1135:
1131:
1127:
1124:
1119:
1115:
1111:
1108:
1105:
1102:
1099:
1096:
1073:
1069:
1065:
1054:
1053:
1042:
1038:
1032:
1029:
1023:
1017:
1014:
1008:
1002:
999:
993:
987:
984:
956:
936:
916:
913:
910:
907:
902:
898:
894:
891:
886:
882:
878:
875:
872:
869:
866:
863:
833:roulette wheel
827:of winning in
737:geometric mean
681:
673:
646:
615:
612:
571:absolute value
541:
538:
535:
513:
510:
507:
503:
498:
492:
487:
484:
481:
477:
473:
468:
464:
443:
442:
431:
428:
425:
403:
400:
397:
393:
388:
385:
380:
376:
358:
357:
346:
341:
338:
335:
331:
324:
320:
316:
311:
307:
292:
291:
280:
275:
272:
269:
265:
260:
257:
252:
248:
228:
213:
210:
189:
188:
177:
171:
166:
162:
158:
152:
147:
143:
139:
133:
128:
124:
120:
114:
111:
108:
102:
99:
26:
9:
6:
4:
3:
2:
3542:
3531:
3528:
3526:
3523:
3522:
3520:
3505:
3497:
3496:
3493:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3454:
3450:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3403:Taylor series
3401:
3400:
3398:
3394:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3335:
3333:
3329:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3304:
3302:
3298:
3295:
3291:
3281:
3278:
3276:
3273:
3271:
3268:
3267:
3265:
3261:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3236:
3234:
3230:
3227:
3223:
3217:
3214:
3212:
3209:
3207:
3204:
3203:
3201:
3197:
3192:
3176:
3173:
3172:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3122:
3120:
3116:
3109:
3103:
3100:
3098:
3095:
3093:
3092:Powers of two
3090:
3088:
3085:
3083:
3080:
3078:
3077:Square number
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3059:
3057:
3053:
3050:
3048:
3044:
3040:
3036:
3029:
3024:
3022:
3017:
3015:
3010:
3009:
3006:
2997:
2996:
2991:
2988:
2983:
2981:
2977:
2974:
2972:
2968:
2965:
2962:
2960:
2959:Mathalino.com
2956:
2953:
2949:
2945:
2944:
2939:
2935:
2934:
2926:
2925:81-8116-000-2
2922:
2918:
2914:
2913:
2904:
2903:
2897:
2891:
2883:
2879:
2875:
2869:
2865:
2861:
2857:
2850:
2843:
2837:
2833:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2781:
2778:
2775:
2772:
2769:
2766:
2765:
2759:
2757:
2756:powers of two
2753:
2752:
2747:
2742:
2740:
2736:
2732:
2728:
2718:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2673:
2670:
2667:
2655:
2651:
2645:
2641:
2632:
2627:
2623:
2615:
2614:
2613:
2600:
2597:
2594:
2591:
2571:
2568:
2565:
2540:
2536:
2503:
2499:
2468:
2463:
2460:
2457:
2446:
2443:
2438:
2434:
2428:
2423:
2419:
2411:
2410:
2409:
2406:
2385:
2378:
2375:
2372:
2366:
2360:
2354:
2351:
2348:
2344:
2340:
2335:
2331:
2323:
2322:
2321:
2297:
2294:
2288:
2285:
2282:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2251:
2245:
2242:
2239:
2235:
2231:
2226:
2222:
2214:
2213:
2212:
2192:
2188:
2184:
2181:
2176:
2173:
2170:
2166:
2162:
2159:
2154:
2150:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2123:
2119:
2111:
2110:
2109:
2095:
2073:
2069:
2054:
2051:
2047:
2042:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2001:
1998:
1995:
1983:
1979:
1973:
1969:
1960:
1955:
1951:
1944:
1941:
1938:
1932:
1928:
1922:
1919:
1916:
1912:
1908:
1903:
1899:
1895:
1890:
1885:
1882:
1879:
1875:
1866:
1853:
1850:
1847:
1839:
1823:
1803:
1794:
1781:
1776:
1772:
1765:
1762:
1759:
1753:
1749:
1743:
1740:
1737:
1733:
1729:
1721:
1714:
1710:
1705:
1700:
1697:
1694:
1690:
1681:
1667:
1650:
1646:
1642:
1639:
1635:
1634:p-adic number
1631:
1627:
1611:
1591:
1583:
1579:
1575:
1558:
1555:
1552:
1545:
1541:
1533:
1517:
1514:
1506:
1478:
1475:
1472:
1465:
1456:
1453:
1450:
1446:
1442:
1439:
1433:
1424:
1411:
1406:
1402:
1398:
1393:
1388:
1385:
1382:
1378:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1347:
1343:
1339:
1336:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1308:
1300:
1281:
1277:
1273:
1263:
1260:
1257:
1253:
1249:
1246:
1243:
1238:
1234:
1230:
1227:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1199:
1192:
1191:
1190:
1188:
1185:The standard
1183:
1181:
1177:
1173:
1155:
1151:
1147:
1144:
1141:
1138:
1133:
1129:
1125:
1122:
1117:
1113:
1109:
1106:
1103:
1100:
1097:
1094:
1085:
1071:
1067:
1063:
1040:
1036:
1030:
1027:
1021:
1015:
1012:
1006:
1000:
997:
991:
985:
982:
973:
972:
971:
970:
954:
934:
914:
911:
908:
905:
900:
896:
892:
889:
884:
880:
876:
873:
870:
867:
864:
861:
852:
850:
846:
842:
838:
834:
830:
826:
825:probabilities
822:
818:
814:
809:
807:
803:
799:
798:Taylor series
795:
791:
787:
783:
779:
775:
771:
767:
766:infinite sums
763:
759:
758:
754:in his work,
753:
748:
746:
742:
738:
734:
730:
726:
718:
714:
710:
706:
702:
697:
685:
678:
665:
656:
645:
640:
634:
630:
624:
620:
611:
609:
605:
601:
600:
595:
591:
586:
584:
580:
576:
572:
567:
565:
561:
555:
552:
539:
536:
533:
511:
508:
505:
501:
496:
490:
485:
482:
479:
475:
471:
466:
462:
453:
450:
448:
429:
426:
423:
401:
398:
395:
391:
386:
383:
378:
374:
366:
365:
364:
363:
344:
339:
336:
333:
329:
322:
318:
314:
309:
305:
297:
296:
295:
278:
273:
270:
267:
263:
258:
255:
250:
246:
238:
237:
236:
234:
227:
223:
219:
209:
207:
206:
200:
198:
194:
175:
169:
164:
160:
156:
150:
145:
141:
137:
131:
126:
122:
118:
112:
109:
106:
100:
97:
90:
89:
88:
86:
82:
78:
75:
70:
68:
64:
60:
57:
53:
49:
41:
37:
32:
19:
3408:Power series
3150:Lucas number
3102:Powers of 10
3082:Cubic number
3066:
2993:
2980:sputsoft.com
2941:
2916:
2901:
2890:
2855:
2849:
2841:
2836:
2750:
2743:
2724:
2716:
2483:
2407:
2404:
2315:
2210:
2060:
2043:
1867:
1795:
1682:
1659:
1425:
1301:
1298:
1184:
1086:
1055:
853:
810:
755:
749:
722:
712:
708:
704:
700:
663:
654:
643:
638:
632:
628:
597:
594:T.R. Malthus
587:
568:
556:
553:
454:
451:
444:
359:
293:
235:is given by
232:
225:
221:
217:
215:
203:
201:
196:
192:
190:
80:
76:
71:
67:common ratio
66:
61:of non-zero
56:mathematical
51:
47:
45:
40:infinite sum
35:
3275:Conditional
3263:Convergence
3254:Telescoping
3239:Alternating
3155:Pell number
1494:, and when
1176:probability
841:investments
725:mathematics
717:unit square
3519:Categories
3300:Convergent
3244:Convergent
2828:References
2484:Rewriting
1180:statistics
969:the series
847:rates and
835:, and the
790:ratio test
762:Archimedes
662:1 −
562:follow an
212:Properties
83:, such as
3331:Divergent
3249:Divergent
3111:Advanced
3087:Factorial
3035:Sequences
2995:MathWorld
2948:EMS Press
2735:Shuruppak
2699:≥
2687:≥
2286:−
2274:⋯
2182:⋅
2174:−
2160:⋯
2144:⋅
2135:⋅
2027:≥
2015:≥
1876:∏
1691:∏
1649:semirings
1556:−
1476:−
1443:−
1379:∑
1356:⋯
1269:∞
1254:∑
1247:⋯
1142:⋯
1041:⋯
845:inflation
627:if |
608:logarithm
509:−
483:−
399:−
337:−
271:−
176:…
3504:Category
3270:Absolute
2967:Archived
2898:(1956).
2762:See also
2751:Elements
2731:Sumerian
1630:function
1172:calculus
927:, where
849:interest
778:calculus
774:parabola
757:Elements
743:are the
606:and the
579:infinity
59:sequence
3280:Uniform
2950:, 2001
2882:2333050
2721:History
1656:Product
1174:and in
851:rates.
711:/ (1 -
668:
651:
647:∞
581:via an
569:If the
63:numbers
54:, is a
3232:Series
3039:series
2923:
2880:
2870:
2746:Euclid
1647:, and
1641:fields
1626:matrix
1534:value
804:, and
770:shapes
752:Euclid
733:series
590:linear
191:where
173:
154:
135:
116:
104:
74:powers
3175:array
3055:Basic
2057:Proof
1796:When
1645:rings
1532:limit
731:is a
3115:list
3037:and
2921:ISBN
2868:ISBN
2595:<
2569:<
2521:and
2061:Let
1816:and
1604:and
1584:for
1578:real
1515:<
1178:and
784:and
727:, a
637:the
537:>
427:>
216:The
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2748:'s
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2525:as
2488:as
1680:is
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1574:.
1084:.
839:of
808:.
723:In
689:1/2
566:.
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2992:.
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1854:1.
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2207:.
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2002:1
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1936:(
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