619:
3175:
31:
3484:
696:
684:
2036:
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
2024:
1043:
2696:
1410:
2189:
1282:
1776:
1854:
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'
2294:
2383:
2463:
1160:
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.
2540:
2503:
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.
3366:
917:
355:
1480:
289:
414:
968:
1516:
2595:
1556:
550:
440:
3356:
2566:
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×
2070:
1848:
3009:
1074:
2090:
1818:
1798:
1662:
1606:
1586:
957:
937:
585:. If the 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.
3449:
1644:
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
446:
960:
3290:
2602:
2959:
2950:
1669:
598:
3300:
2855:
1292:
2938:
2713:(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
2098:
1183:
3295:
2710:
3464:
3055:
3002:
2774:
719:
is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.
3444:
3346:
3336:
2908:
1824:
of the partial progression's first and last individual terms and then raising that mean to the power given by the number of terms
3454:
93:
2757:
2201:
3508:
3459:
3361:
2995:
2931:
2780:
3513:
3487:
2310:
3469:
2926:
2398:
1082:
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.
2019:{\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
3341:
3331:
3321:
3351:
2763:
589:
803:
Geometric series have been applied to model a wide variety of natural phenomena and social phenomena, such as the
2842:. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153.
804:
577:. If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approach
2512:
2475:
457:
3436:
3258:
849:
618:
2921:
300:
3098:
3045:
2838:
Friberg, Jöran (2007). "MS 3047: An Old
Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.).
2804:
2751:
2722:
1613:
563:
3305:
2947:
2798:
1417:
593:
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.
3416:
3253:
3022:
2786:
2768:
2733:
1609:
756:
732:
603:
573:
of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via an
1162:
are called "finite geometric series" in certain branches of mathematics, especially in 19th century
1038:{\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }
3396:
3263:
2942:
2795: – Standard guidelines for choosing exact product dimensions within a given set of constraints
2029:
1624:
622:
361:
2963:
811:, the decay of radioactive carbon-14 atoms where the common ratio between terms is defined by the
3237:
3222:
3194:
3174:
3113:
1628:
812:
797:
582:
3326:
3426:
3227:
3199:
3153:
3143:
3123:
3108:
2970:
2904:
2851:
1485:
1048:
is geometric because each successive term can be obtained by multiplying the previous term by
828:
781:
740:
574:
38:) up to 6 iterations deep. The first block is a unit block and the dashed line represents the
2571:
1521:
529:
419:
3411:
3232:
3158:
3148:
3128:
3030:
2843:
2792:
2545:
1621:
785:
728:
676:
559:
204:
2865:
2048:
3189:
3118:
2954:
2861:
2033:
1827:
744:
1051:
808:
3421:
3406:
3401:
3080:
3065:
2075:
1821:
1803:
1783:
1647:
1591:
1571:
1565:
942:
922:
824:
820:
793:
736:
570:
73:
3502:
3386:
3060:
2973:
2884:
2739:
2721:. It is the only known record of a geometric progression from before the time of old
1617:
789:
3391:
3133:
3075:
765:
695:
84:
30:
17:
2777: – Progression formed by taking the reciprocals of an arithmetic progression
3138:
3085:
2879:
1561:
1167:
816:
724:
716:
55:
2691:{\displaystyle P_{n}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0}
683:
2847:
1171:
832:
761:
2304:
is the sum of an arithmetic sequence. Substituting the formula for that sum,
1177:
The standard generator form expression for the infinite geometric series is
3070:
2978:
2718:
836:
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
607:
2939:
Derivation of formulas for sum of finite and infinite geometric progression
788:. They serve as prototypes for frequently used mathematical tools such as
3018:
1632:
1163:
840:
777:
773:
578:
58:
554:
This is a second order nonlinear recurrence with constant coefficients.
2987:
1620:
geometric series, and, most generally, geometric series of elements of
1405:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}
2184:{\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 =
34:
Diagram illustrating three basic geometric sequences of the pattern 1(
2742:, see the article for details) and give several of their properties.
2729:
2032:: the sum of an arithmetic sequence is the number of terms times the
1560:
Though geometric series are most commonly found and applied with the
1287:
and the generator form expression for the finite geometric series is
1277:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots =\sum _{k=0}^{\infty }ar^{k}}
751:
62:
2028:
This corresponds to a similar property of sums of terms of a finite
452:
Geometric sequences also satisfy the nonlinear recurrence relation
2714:
769:
694:
682:
617:
29:
2991:
835:
where the common ratio could be determined by a combination of
739:
of its two neighbouring terms, similar to how the terms in an
1771:{\displaystyle \prod _{k=0}^{n}ar^{(k)}=a^{n+1}r^{n(n+1)/2}.}
768:, particularly in calculating areas and volumes of geometric
1820:
are positive real numbers, this is equivalent to taking the
823:
where the common ratio could be determined by the odds of a
2760: – Mathematical sequence satisfying a specific pattern
2195:
Carrying out the multiplications and gathering like terms,
181:{\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }
959:
is the common ratio between adjacent terms. For example,
625:
of the formula for the sum of a geometric series –
2840:
A remarkable collection of
Babylonian mathematical texts
2828:
Belmont, California, Wadsworth
Publishing, p. 566, 1970.
1608:, 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
2516:
2479:
2289:{\displaystyle P_{n}=a^{n+1}r^{1+2+3+\cdots +(n-1)+n}}
3367:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
3357:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
2701:
which is the formula in terms of the geometric mean.
2605:
2574:
2548:
2515:
2478:
2401:
2313:
2204:
2101:
2078:
2051:
1857:
1830:
1806:
1786:
1672:
1650:
1594:
1574:
1524:
1488:
1420:
1295:
1186:
1085:
1054:
971:
945:
925:
852:
780:, where they have been paradigmatic examples of both
532:
460:
422:
372:
303:
244:
96:
2783: – Divergent sum of all positive unit fractions
2771: – Mathematical function, denoted exp(x) or e^x
3435:
3379:
3314:
3283:
3276:
3246:
3215:
3208:
3182:
3094:
3038:
3029:
807: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
2890:(2nd ed. ed.). New York: Dover Publications.
2883:
2690:
2589:
2560:
2534:
2497:
2457:
2378:{\displaystyle P_{n}=a^{n+1}r^{\frac {n(n+1)}{2}}}
2377:
2288:
2183:
2084:
2064:
2018:
1842:
1812:
1792:
1770:
1656:
1600:
1580:
1550:
1510:
1474:
1404:
1276:
1154:
1068:
1037:
951:
931:
911:
544:
518:
434:
408:
349:
283:
180:
2458:{\displaystyle P_{n}=(ar^{\frac {n}{2}})^{n+1}.}
1155:{\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}}
2801: – British political economist (1766–1834)
764:further advanced the study through his work on
2826:Calculus and Analytic Geometry, Second Edition
3003:
846:In general, a geometric series is written as
8:
3450:Hypergeometric function of a matrix argument
2738:analyze geometric progressions (such as the
772:(for instance calculating the area inside a
447:linear recurrence with constant coefficients
3306:1 + 1/2 + 1/3 + ... (Riemann zeta function)
1518:the infinite series converges to the value
3280:
3212:
3035:
3010:
2996:
2988:
2754: – Sequence of equally spaced numbers
2535:{\displaystyle \textstyle {\sqrt {r^{2}}}}
2498:{\displaystyle \textstyle {\sqrt {a^{2}}}}
3362:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
2960:Nice Proof of a Geometric Progression Sum
2662:
2650:
2638:
2628:
2622:
2610:
2604:
2573:
2547:
2523:
2517:
2514:
2486:
2480:
2477:
2440:
2425:
2406:
2400:
2347:
2331:
2318:
2312:
2238:
2222:
2209:
2203:
2175:
2153:
2137:
2106:
2100:
2077:
2056:
2050:
1990:
1978:
1966:
1956:
1950:
1934:
1915:
1899:
1886:
1873:
1862:
1856:
1829:
1805:
1785:
1755:
1736:
1720:
1701:
1688:
1677:
1671:
1649:
1593:
1573:
1528:
1523:
1497:
1489:
1487:
1452:
1437:
1419:
1393:
1380:
1369:
1356:
1334:
1318:
1294:
1268:
1255:
1244:
1225:
1209:
1185:
1146:
1124:
1108:
1084:
1058:
1053:
1031:
1027:
1017:
1016:
1012:
1002:
1001:
997:
987:
986:
982:
972:
970:
944:
924:
891:
875:
851:
531:
519:{\displaystyle a_{n}=a_{n-1}^{2}/a_{n-2}}
504:
495:
489:
478:
465:
459:
421:
394:
389:
377:
371:
332:
327:
321:
308:
302:
266:
261:
249:
243:
163:
144:
125:
95:
1414:Any finite geometric series has the sum
760:, which explored geometric proportions.
2886:The Thirteen Books of Euclid's Elements
2817:
599:An Essay on the Principle of Population
360:Geometric sequences satisfy the linear
912:{\displaystyle a+ar+ar^{2}+ar^{3}+...}
699:Another geometric series (coefficient
596:as the mathematical foundation of his
2392:One can rearrange this expression to
2037:to products of exponentiated values.
350:{\displaystyle a_{n}=a_{m}\,r^{n-m}.}
72:Examples of a geometric sequence are
7:
2711:Early Dynastic Period in Mesopotamia
3327:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
445:This is a first order, homogeneous
2072:represent the product up to power
1475:{\displaystyle a(1-r^{n+1})/(1-r)}
1256:
747:of their two neighbouring terms.
25:
3445:Generalized hypergeometric series
284:{\displaystyle a_{n}=a\,r^{n-1},}
3483:
3482:
3455:Lauricella hypergeometric series
3173:
2948:Geometric Progression Calculator
2807: – Probability distribution
675:This section is an excerpt from
409:{\displaystyle a_{n}=r\,a_{n-1}}
27:Mathematical sequence of numbers
3465:Riemann's differential equation
776:) and the early development of
2758:Arithmetico-geometric sequence
2647:
2619:
2437:
2415:
2365:
2353:
2275:
2263:
1975:
1947:
1931:
1919:
1752:
1740:
1708:
1702:
1545:
1533:
1498:
1490:
1469:
1457:
1449:
1424:
1:
3460:Modular hypergeometric series
3301:1/4 + 1/16 + 1/64 + 1/256 + ⋯
2542:though this is not valid for
2389:which concludes the proof.
3470:Theta hypergeometric series
2927:Encyclopedia of Mathematics
1079:Truncated geometric series
79:of a fixed non-zero number
39:
3530:
3352:Infinite arithmetic series
3296:1/2 + 1/4 + 1/8 + 1/16 + ⋯
3291:1/2 − 1/4 + 1/8 − 1/16 + ⋯
2764:Linear difference equation
1616:-valued geometric series,
1612:-valued geometric series,
674:
3478:
3171:
2848:10.1007/978-0-387-48977-3
805:expansion of the universe
1511:{\displaystyle |r|<1}
1174:and their applications.
939:is the initial term and
195:is the common ratio and
3183:Properties of sequences
2922:"Geometric progression"
2709:A clay tablet from the
2590:{\displaystyle r<0,}
2092:. Written out in full,
1551:{\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.
3046:Arithmetic progression
2805:Geometric distribution
2752:Arithmetic progression
2725:beginning in 2000 BC.
2723:Babylonian mathematics
2692:
2591:
2562:
2561:{\displaystyle a<0}
2536:
2499:
2459:
2379:
2290:
2185:
2086:
2066:
2020:
1878:
1844:
1814:
1794:
1772:
1693:
1658:
1602:
1582:
1552:
1512:
1476:
1406:
1385:
1278:
1260:
1156:
1070:
1039:
953:
933:
913:
813:half-life of carbon-14
720:
692:
671:
564:arithmetic progression
546:
520:
436:
410:
351:
285:
182:
43:
3437:Hypergeometric series
3051:Geometric progression
2799:Thomas Robert Malthus
2728:Books VIII and IX of
2693:
2592:
2563:
2537:
2500:
2460:
2380:
2291:
2186:
2087:
2067:
2065:{\displaystyle P_{n}}
2021:
1858:
1845:
1815:
1795:
1773:
1673:
1659:
1603:
1583:
1553:
1513:
1477:
1407:
1365:
1279:
1240:
1157:
1071:
1040:
954:
934:
914:
698:
686:
621:
547:
521:
437:
411:
352:
286:
183:
48:geometric progression
33:
3509:Sequences and series
3417:Trigonometric series
3209:Properties of series
3056:Harmonic progression
2789: – Infinite sum
2775:Harmonic progression
2769:Exponential function
2603:
2572:
2546:
2513:
2476:
2399:
2311:
2202:
2099:
2076:
2049:
1855:
1843:{\displaystyle n+1.}
1828:
1804:
1784:
1670:
1648:
1592:
1572:
1522:
1486:
1418:
1293:
1184:
1083:
1052:
969:
943:
923:
850:
604:exponential function
530:
458:
420:
370:
301:
242:
94:
3514:Mathematical series
3397:Formal power series
2899:Hall & Knight,
2824:Riddle, Douglas F.
2717:, from the city of
2030:arithmetic sequence
1069:{\displaystyle 1/2}
798:matrix exponentials
623:Proof without words
494:
362:recurrence relation
3195:Monotonic function
3114:Fibonacci sequence
2974:"Geometric Series"
2971:Weisstein, Eric W.
2953:2008-12-27 at the
2688:
2587:
2558:
2532:
2531:
2495:
2494:
2455:
2375:
2286:
2181:
2082:
2062:
2016:
1840:
1810:
1790:
1768:
1654:
1622:abstract algebraic
1598:
1578:
1548:
1508:
1472:
1402:
1274:
1152:
1066:
1035:
949:
929:
909:
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
3496:
3495:
3427:Generating series
3375:
3374:
3347:1 − 2 + 4 − 8 + ⋯
3342:1 + 2 + 4 + 8 + ⋯
3337:1 − 2 + 3 − 4 + ⋯
3332:1 + 2 + 3 + 4 + ⋯
3322:1 + 1 + 1 + 1 + ⋯
3272:
3271:
3200:Periodic sequence
3169:
3168:
3154:Triangular number
3144:Pentagonal number
3124:Heptagonal number
3109:Complete sequence
3031:Integer sequences
2857:978-0-387-34543-7
2665:
2644:
2529:
2492:
2433:
2372:
2085:{\displaystyle n}
1993:
1972:
1813:{\displaystyle r}
1793:{\displaystyle a}
1657:{\displaystyle n}
1601:{\displaystyle r}
1581:{\displaystyle a}
1025:
1010:
995:
980:
952:{\displaystyle r}
932:{\displaystyle a}
809: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
3521:
3486:
3485:
3412:Dirichlet series
3281:
3213:
3177:
3149:Polygonal number
3129:Hexagonal number
3102:
3036:
3012:
3005:
2998:
2989:
2984:
2983:
2935:
2892:
2891:
2889:
2880:Heath, Thomas L.
2876:
2870:
2869:
2835:
2829:
2822:
2793:Preferred number
2697:
2695:
2694:
2689:
2666:
2663:
2661:
2660:
2645:
2643:
2642:
2633:
2632:
2623:
2615:
2614:
2596:
2594:
2593:
2588:
2567:
2565:
2564:
2559:
2541:
2539:
2538:
2533:
2530:
2528:
2527:
2518:
2508:
2504:
2502:
2501:
2496:
2493:
2491:
2490:
2481:
2471:
2464:
2462:
2461:
2456:
2451:
2450:
2435:
2434:
2426:
2411:
2410:
2384:
2382:
2381:
2376:
2374:
2373:
2368:
2348:
2342:
2341:
2323:
2322:
2303:
2300:The exponent of
2295:
2293:
2292:
2287:
2285:
2284:
2233:
2232:
2214:
2213:
2190:
2188:
2187:
2182:
2180:
2179:
2164:
2163:
2142:
2141:
2111:
2110:
2091:
2089:
2088:
2083:
2071:
2069:
2068:
2063:
2061:
2060:
2025:
2023:
2022:
2017:
1994:
1991:
1989:
1988:
1973:
1971:
1970:
1961:
1960:
1951:
1943:
1942:
1938:
1910:
1909:
1891:
1890:
1877:
1872:
1849:
1847:
1846:
1841:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1777:
1775:
1774:
1769:
1764:
1763:
1759:
1731:
1730:
1712:
1711:
1692:
1687:
1663:
1661:
1660:
1655:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1557:
1555:
1554:
1549:
1532:
1517:
1515:
1514:
1509:
1501:
1493:
1481:
1479:
1478:
1473:
1456:
1448:
1447:
1411:
1409:
1408:
1403:
1398:
1397:
1384:
1379:
1361:
1360:
1339:
1338:
1323:
1322:
1283:
1281:
1280:
1275:
1273:
1272:
1259:
1254:
1230:
1229:
1214:
1213:
1161:
1159:
1158:
1153:
1151:
1150:
1129:
1128:
1113:
1112:
1075:
1073:
1072:
1067:
1062:
1044:
1042:
1041:
1036:
1026:
1018:
1011:
1003:
996:
988:
981:
973:
958:
956:
955:
950:
938:
936:
935:
930:
918:
916:
915:
910:
896:
895:
880:
879:
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:
3529:
3528:
3524:
3523:
3522:
3520:
3519:
3518:
3499:
3498:
3497:
3492:
3474:
3431:
3380:Kinds of series
3371:
3310:
3277:Explicit series
3268:
3242:
3204:
3190:Cauchy sequence
3178:
3165:
3119:Figurate number
3096:
3090:
3081:Powers of three
3025:
3016:
2969:
2968:
2955:Wayback Machine
2920:
2917:
2896:
2895:
2878:
2877:
2873:
2858:
2837:
2836:
2832:
2823:
2819:
2814:
2787:Infinite series
2781:Harmonic series
2748:
2707:
2664: for
2646:
2634:
2624:
2606:
2601:
2600:
2570:
2569:
2544:
2543:
2519:
2511:
2510:
2506:
2482:
2474:
2473:
2469:
2436:
2421:
2402:
2397:
2396:
2349:
2343:
2327:
2314:
2309:
2308:
2301:
2234:
2218:
2205:
2200:
2199:
2171:
2149:
2133:
2102:
2097:
2096:
2074:
2073:
2052:
2047:
2046:
2043:
2034:arithmetic mean
1992: for
1974:
1962:
1952:
1911:
1895:
1882:
1853:
1852:
1826:
1825:
1802:
1801:
1782:
1781:
1732:
1716:
1697:
1668:
1667:
1646:
1645:
1642:
1637:
1636:
1590:
1589:
1570:
1569:
1566:complex numbers
1520:
1519:
1484:
1483:
1433:
1416:
1415:
1389:
1352:
1330:
1314:
1291:
1290:
1264:
1221:
1205:
1182:
1181:
1142:
1120:
1104:
1081:
1080:
1050:
1049:
967:
966:
941:
940:
921:
920:
887:
871:
848:
847:
829:economic values
821:games of chance
688:
680:
661:
658:
653:
652:
650:
648:
642:
626:
616:
528:
527:
500:
461:
456:
455:
418:
417:
390:
373:
368:
367:
328:
317:
304:
299:
298:
294:and in general
262:
245:
240:
239:
230:
214:
159:
140:
121:
92:
91:
28:
23:
22:
15:
12:
11:
5:
3527:
3525:
3517:
3516:
3511:
3501:
3500:
3494:
3493:
3491:
3490:
3479:
3476:
3475:
3473:
3472:
3467:
3462:
3457:
3452:
3447:
3441:
3439:
3433:
3432:
3430:
3429:
3424:
3422:Fourier series
3419:
3414:
3409:
3407:Puiseux series
3404:
3402:Laurent series
3399:
3394:
3389:
3383:
3381:
3377:
3376:
3373:
3372:
3370:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3318:
3316:
3312:
3311:
3309:
3308:
3303:
3298:
3293:
3287:
3285:
3278:
3274:
3273:
3270:
3269:
3267:
3266:
3261:
3256:
3250:
3248:
3244:
3243:
3241:
3240:
3235:
3230:
3225:
3219:
3217:
3210:
3206:
3205:
3203:
3202:
3197:
3192:
3186:
3184:
3180:
3179:
3172:
3170:
3167:
3166:
3164:
3163:
3162:
3161:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3105:
3103:
3092:
3091:
3089:
3088:
3083:
3078:
3073:
3068:
3063:
3058:
3053:
3048:
3042:
3040:
3033:
3027:
3026:
3017:
3015:
3014:
3007:
3000:
2992:
2986:
2985:
2966:
2957:
2945:
2936:
2916:
2915:External links
2913:
2912:
2911:
2903:, p. 39,
2901:Higher Algebra
2894:
2893:
2871:
2856:
2830:
2816:
2815:
2813:
2810:
2809:
2808:
2802:
2796:
2790:
2784:
2778:
2772:
2766:
2761:
2755:
2747:
2744:
2706:
2703:
2699:
2698:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2659:
2656:
2653:
2649:
2641:
2637:
2631:
2627:
2621:
2618:
2613:
2609:
2586:
2583:
2580:
2577:
2557:
2554:
2551:
2526:
2522:
2489:
2485:
2466:
2465:
2454:
2449:
2446:
2443:
2439:
2432:
2429:
2424:
2420:
2417:
2414:
2409:
2405:
2387:
2386:
2371:
2367:
2364:
2361:
2358:
2355:
2352:
2346:
2340:
2337:
2334:
2330:
2326:
2321:
2317:
2298:
2297:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2237:
2231:
2228:
2225:
2221:
2217:
2212:
2208:
2193:
2192:
2178:
2174:
2170:
2167:
2162:
2159:
2156:
2152:
2148:
2145:
2140:
2136:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2109:
2105:
2081:
2059:
2055:
2042:
2039:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1987:
1984:
1981:
1977:
1969:
1965:
1959:
1955:
1949:
1946:
1941:
1937:
1933:
1930:
1927:
1924:
1921:
1918:
1914:
1908:
1905:
1902:
1898:
1894:
1889:
1885:
1881:
1876:
1871:
1868:
1865:
1861:
1839:
1836:
1833:
1822:geometric mean
1809:
1789:
1767:
1762:
1758:
1754:
1751:
1748:
1745:
1742:
1739:
1735:
1729:
1726:
1723:
1719:
1715:
1710:
1707:
1704:
1700:
1696:
1691:
1686:
1683:
1680:
1676:
1653:
1641:
1638:
1597:
1577:
1547:
1544:
1541:
1538:
1535:
1531:
1527:
1507:
1504:
1500:
1496:
1492:
1471:
1468:
1465:
1462:
1459:
1455:
1451:
1446:
1443:
1440:
1436:
1432:
1429:
1426:
1423:
1401:
1396:
1392:
1388:
1383:
1378:
1375:
1372:
1368:
1364:
1359:
1355:
1351:
1348:
1345:
1342:
1337:
1333:
1329:
1326:
1321:
1317:
1313:
1310:
1307:
1304:
1301:
1298:
1285:
1284:
1271:
1267:
1263:
1258:
1253:
1250:
1247:
1243:
1239:
1236:
1233:
1228:
1224:
1220:
1217:
1212:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1149:
1145:
1141:
1138:
1135:
1132:
1127:
1123:
1119:
1116:
1111:
1107:
1103:
1100:
1097:
1094:
1091:
1088:
1065:
1061:
1057:
1046:
1045:
1034:
1030:
1024:
1021:
1015:
1009:
1006:
1000:
994:
991:
985:
979:
976:
948:
928:
908:
905:
902:
899:
894:
890:
886:
883:
878:
874:
870:
867:
864:
861:
858:
855:
825:roulette wheel
819:of winning in
794:Fourier series
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3526:
3515:
3512:
3510:
3507:
3506:
3504:
3489:
3481:
3480:
3477:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3442:
3440:
3438:
3434:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3387:Taylor series
3385:
3384:
3382:
3378:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3319:
3317:
3313:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3286:
3282:
3279:
3275:
3265:
3262:
3260:
3257:
3255:
3252:
3251:
3249:
3245:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3220:
3218:
3214:
3211:
3207:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3185:
3181:
3176:
3160:
3157:
3156:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3106:
3104:
3100:
3093:
3087:
3084:
3082:
3079:
3077:
3076:Powers of two
3074:
3072:
3069:
3067:
3064:
3062:
3061:Square number
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3043:
3041:
3037:
3034:
3032:
3028:
3024:
3020:
3013:
3008:
3006:
3001:
2999:
2994:
2993:
2990:
2981:
2980:
2975:
2972:
2967:
2965:
2961:
2958:
2956:
2952:
2949:
2946:
2944:
2943:Mathalino.com
2940:
2937:
2933:
2929:
2928:
2923:
2919:
2918:
2914:
2910:
2909:81-8116-000-2
2906:
2902:
2898:
2897:
2888:
2887:
2881:
2875:
2872:
2867:
2863:
2859:
2853:
2849:
2845:
2841:
2834:
2831:
2827:
2821:
2818:
2811:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2765:
2762:
2759:
2756:
2753:
2750:
2749:
2745:
2743:
2741:
2740:powers of two
2737:
2736:
2731:
2726:
2724:
2720:
2716:
2712:
2704:
2702:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2657:
2654:
2651:
2639:
2635:
2629:
2625:
2616:
2611:
2607:
2599:
2598:
2597:
2584:
2581:
2578:
2575:
2555:
2552:
2549:
2524:
2520:
2487:
2483:
2452:
2447:
2444:
2441:
2430:
2427:
2422:
2418:
2412:
2407:
2403:
2395:
2394:
2393:
2390:
2369:
2362:
2359:
2356:
2350:
2344:
2338:
2335:
2332:
2328:
2324:
2319:
2315:
2307:
2306:
2305:
2281:
2278:
2272:
2269:
2266:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2235:
2229:
2226:
2223:
2219:
2215:
2210:
2206:
2198:
2197:
2196:
2176:
2172:
2168:
2165:
2160:
2157:
2154:
2150:
2146:
2143:
2138:
2134:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2107:
2103:
2095:
2094:
2093:
2079:
2057:
2053:
2040:
2038:
2035:
2031:
2026:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1985:
1982:
1979:
1967:
1963:
1957:
1953:
1944:
1939:
1935:
1928:
1925:
1922:
1916:
1912:
1906:
1903:
1900:
1896:
1892:
1887:
1883:
1879:
1874:
1869:
1866:
1863:
1859:
1850:
1837:
1834:
1831:
1823:
1807:
1787:
1778:
1765:
1760:
1756:
1749:
1746:
1743:
1737:
1733:
1727:
1724:
1721:
1717:
1713:
1705:
1698:
1694:
1689:
1684:
1681:
1678:
1674:
1665:
1651:
1639:
1634:
1630:
1626:
1623:
1619:
1618:p-adic number
1615:
1611:
1595:
1575:
1567:
1563:
1559:
1542:
1539:
1536:
1529:
1525:
1505:
1502:
1494:
1466:
1463:
1460:
1453:
1444:
1441:
1438:
1434:
1430:
1427:
1421:
1412:
1399:
1394:
1390:
1386:
1381:
1376:
1373:
1370:
1366:
1362:
1357:
1353:
1349:
1346:
1343:
1340:
1335:
1331:
1327:
1324:
1319:
1315:
1311:
1308:
1305:
1302:
1299:
1296:
1288:
1269:
1265:
1261:
1251:
1248:
1245:
1241:
1237:
1234:
1231:
1226:
1222:
1218:
1215:
1210:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1180:
1179:
1178:
1175:
1173:
1169:
1165:
1147:
1143:
1139:
1136:
1133:
1130:
1125:
1121:
1117:
1114:
1109:
1105:
1101:
1098:
1095:
1092:
1089:
1086:
1077:
1063:
1059:
1055:
1032:
1028:
1022:
1019:
1013:
1007:
1004:
998:
992:
989:
983:
977:
974:
965:
964:
963:
962:
946:
926:
906:
903:
900:
897:
892:
888:
884:
881:
876:
872:
868:
865:
862:
859:
856:
853:
844:
842:
838:
834:
830:
826:
822:
818:
817:probabilities
814:
810:
806:
801:
799:
795:
791:
790:Taylor series
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:
613:
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:
211:
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:
3392:Power series
3134:Lucas number
3086:Powers of 10
3066:Cubic number
3050:
2977:
2964:sputsoft.com
2925:
2900:
2885:
2874:
2839:
2833:
2825:
2820:
2734:
2727:
2708:
2700:
2467:
2391:
2388:
2299:
2194:
2044:
2027:
1851:
1779:
1666:
1643:
1413:
1289:
1286:
1176:
1078:
1047:
845:
802:
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:
3259:Conditional
3247:Convergence
3238:Telescoping
3223:Alternating
3139:Pell number
1482:, and when
1168:probability
833:investments
725:mathematics
717:unit square
3503:Categories
3284:Convergent
3228:Convergent
2812:References
2468:Rewriting
1172:statistics
961:the series
839:rates and
827:, and the
762:Archimedes
662:1 −
562:follow an
212:Properties
83:, such as
3315:Divergent
3233:Divergent
3095:Advanced
3071:Factorial
3019:Sequences
2979:MathWorld
2932:EMS Press
2719:Shuruppak
2683:≥
2671:≥
2270:−
2258:⋯
2166:⋅
2158:−
2144:⋯
2128:⋅
2119:⋅
2011:≥
1999:≥
1860:∏
1675:∏
1633:semirings
1540:−
1464:−
1431:−
1367:∑
1344:⋯
1257:∞
1242:∑
1235:⋯
1134:⋯
1033:⋯
837:inflation
627:if |
608:logarithm
509:−
483:−
399:−
337:−
271:−
176:…
3488:Category
3254:Absolute
2951:Archived
2882:(1956).
2746:See also
2735:Elements
2715:Sumerian
1614:function
1164:calculus
919:, where
841:interest
778:calculus
774:parabola
757:Elements
743:are the
606:and the
579:infinity
59:sequence
3264:Uniform
2934:, 2001
2866:2333050
2705:History
1640:Product
1166:and in
843:rates.
711:/ (1 -
668:
651:
647:∞
581:via an
569:If the
63:numbers
54:, is a
3216:Series
3023:series
2907:
2864:
2854:
2730:Euclid
1631:, and
1625:fields
1610:matrix
796:, and
770:shapes
752:Euclid
733:series
590:linear
191:where
173:
154:
135:
116:
104:
74:powers
3159:array
3039:Basic
2041:Proof
1780:When
1629:rings
731:is a
3099:list
3021:and
2905:ISBN
2852:ISBN
2579:<
2553:<
2505:and
2045:Let
1800:and
1588:and
1568:for
1562:real
1503:<
1170:and
784:and
727:, a
637:the
537:>
427:>
216:The
2962:at
2941:at
2844:doi
2732:'s
2568:or
2509:as
2472:as
1664:is
1564:or
1558:.
1076:.
831:of
800:.
723:In
689:1/2
566:.
3505::
2976:.
2930:,
2924:,
2862:MR
2860:.
2850:.
2014:0.
1838:1.
1627:,
1023:16
815:,
792:,
649:=
540:2.
449:.
430:1.
224:=
208:.
46:A
3101:)
3097:(
3011:e
3004:t
2997:v
2982:.
2868:.
2846::
2686:0
2680:r
2677:,
2674:0
2668:a
2658:1
2655:+
2652:n
2648:)
2640:n
2636:r
2630:2
2626:a
2620:(
2617:=
2612:n
2608:P
2585:,
2582:0
2576:r
2556:0
2550:a
2525:2
2521:r
2507:r
2488:2
2484:a
2470:a
2453:.
2448:1
2445:+
2442:n
2438:)
2431:2
2428:n
2423:r
2419:a
2416:(
2413:=
2408:n
2404:P
2385:,
2370:2
2366:)
2363:1
2360:+
2357:n
2354:(
2351:n
2345:r
2339:1
2336:+
2333:n
2329:a
2325:=
2320:n
2316:P
2302:r
2296:.
2282:n
2279:+
2276:)
2273:1
2267:n
2264:(
2261:+
2255:+
2252:3
2249:+
2246:2
2243:+
2240:1
2236:r
2230:1
2227:+
2224:n
2220:a
2216:=
2211:n
2207:P
2191:.
2177:n
2173:r
2169:a
2161:1
2155:n
2151:r
2147:a
2139:2
2135:r
2131:a
2125:r
2122:a
2116:a
2113:=
2108:n
2104:P
2080:n
2058:n
2054:P
2008:r
2005:,
2002:0
1996:a
1986:1
1983:+
1980:n
1976:)
1968:n
1964:r
1958:2
1954:a
1948:(
1945:=
1940:2
1936:/
1932:)
1929:1
1926:+
1923:n
1920:(
1917:n
1913:r
1907:1
1904:+
1901:n
1897:a
1893:=
1888:k
1884:r
1880:a
1875:n
1870:0
1867:=
1864:k
1835:+
1832:n
1808:r
1788:a
1766:.
1761:2
1757:/
1753:)
1750:1
1747:+
1744:n
1741:(
1738:n
1734:r
1728:1
1725:+
1722:n
1718:a
1714:=
1709:)
1706:k
1703:(
1699:r
1695:a
1690:n
1685:0
1682:=
1679:k
1652:n
1635:.
1596:r
1576:a
1546:)
1543:r
1537:1
1534:(
1530:/
1526:a
1506:1
1499:|
1495:r
1491:|
1470:)
1467:r
1461:1
1458:(
1454:/
1450:)
1445:1
1442:+
1439:n
1435:r
1428:1
1425:(
1422:a
1400:.
1395:k
1391:r
1387:a
1382:n
1377:0
1374:=
1371:k
1363:=
1358:n
1354:r
1350:a
1347:+
1341:+
1336:3
1332:r
1328:a
1325:+
1320:2
1316:r
1312:a
1309:+
1306:r
1303:a
1300:+
1297:a
1270:k
1266:r
1262:a
1252:0
1249:=
1246:k
1238:=
1232:+
1227:3
1223:r
1219:a
1216:+
1211:2
1207:r
1203:a
1200:+
1197:r
1194:a
1191:+
1188:a
1148:n
1144:r
1140:a
1137:+
1131:+
1126:3
1122:r
1118:a
1115:+
1110:2
1106:r
1102:a
1099:+
1096:r
1093:a
1090:+
1087:a
1064:2
1060:/
1056:1
1029:+
1020:1
1014:+
1008:8
1005:1
999:+
993:4
990:1
984:+
978:2
975:1
947:r
927:a
907:.
904:.
901:.
898:+
893:3
889:r
885:a
882:+
877:2
873:r
869:a
866:+
863:r
860:a
857:+
854:a
713:r
709:a
705:r
701:a
679:.
664:r
659:/
655:a
644:S
639:r
633:n
629:r
534:n
512:2
506:n
502:a
497:/
491:2
486:1
480:n
476:a
472:=
467:n
463:a
424:n
402:1
396:n
392:a
387:r
384:=
379:n
375:a
345:.
340:m
334:n
330:r
323:m
319:a
315:=
310:n
306:a
279:,
274:1
268:n
264:r
259:a
256:=
251:n
247:a
233:r
229:1
226:a
222:a
218:n
197:a
193:r
170:,
165:4
161:r
157:a
151:,
146:3
142:r
138:a
132:,
127:2
123:r
119:a
113:,
110:r
107:a
101:,
98:a
85:2
81:r
77:r
36:r
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
