2226:
20:
2535:
1122:
1386:
The above manipulation might be called on to produce −1 outside the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A
972:
shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid:
979:
513:
866:
261:
1555:
314:
687:
177:
631:
1767:
1587:
1467:
1429:
970:
772:
373:
130:
544:
1251:
381:
1518:
1494:
1117:{\displaystyle {\begin{array}{rcl}s&=&\displaystyle 1+2+4+8+16+\cdots \\&=&\displaystyle 1+2(1+2+4+8+\cdots )\\&=&\displaystyle 1+2s\end{array}}}
1151:
1298:
728:
1209:
1189:
1969:
1714:
1353:
1382:
924:
892:
570:
1318:
1275:
590:
2417:
1863:
1609:
2060:
792:
2407:
2500:
182:
789:
An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is,
2341:
2351:
1980:
1600:. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.
2515:
2346:
2106:
2053:
1907:
Ferraro, Giovanni (2002). "Convergence and Formal
Manipulation of Series from the Origins of Calculus to About 1730".
1596:
It is also possible to view this series as convergent in a number system different from the real numbers, namely, the
2495:
1841:
1788:
2505:
1829:
320:
2397:
2387:
1634:
1624:
1524:
271:
2510:
2412:
2046:
636:
2538:
1469:
These examples illustrate the potential danger in applying similar arguments to the series implied by such
68:
However, it can be manipulated to yield a number of mathematically interesting results. For example, many
2520:
2559:
2402:
135:
603:
2569:
2564:
2382:
2372:
1728:
1629:
1619:
1560:
1442:
1390:
1212:
931:
733:
334:
91:
522:
1783:. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. pp. chapter I, exercise 16, p. 20.
1221:
1500:
1476:
1216:
50:
2574:
2487:
2309:
72:
are used in mathematics to assign numerical values even to a divergent series. For example, the
2149:
2096:
1130:
1280:
692:
2356:
2101:
1194:
1156:
593:
516:
1684:
1642:, a data convention for representing negative numbers where −1 is represented as if it were
1323:
2467:
2304:
2073:
1358:
897:
871:
549:
8:
2447:
2314:
1639:
73:
2001:(November 1987). "Humanities students and epistemological obstacles related to limits".
324:
2377:
2288:
2273:
2245:
2225:
2164:
2026:
2018:
1956:
1924:
1813:
1614:
1590:
1432:
1303:
1260:
575:
1998:
984:
508:{\displaystyle f(x)=1+2x+4x^{2}+8x^{3}+\cdots +2^{n}{}x^{n}+\cdots ={\frac {1}{1-2x}}}
2477:
2278:
2250:
2204:
2194:
2174:
2159:
2030:
1928:
1898:
1881:
1837:
1784:
1470:
1521:. The arguments are ultimately justified for these convergent series, implying that
2462:
2283:
2209:
2199:
2179:
2081:
2010:
1948:
1916:
1893:
1859:
69:
58:
46:
2240:
2169:
775:
38:
19:
2472:
2457:
2452:
2131:
2116:
1809:
1254:
783:
328:
77:
1920:
2553:
2437:
2111:
1597:
597:
331:. On the other hand, there is at least one generally useful method that sums
42:
2442:
2184:
2126:
1936:
376:
2189:
2136:
1851:
779:
54:
28:
2038:
2022:
2014:
1960:
1257:). If some summation method is known to return an ordinary number for
778:) to −1, and −1 is the (E) sum of the series. (The notation is due to
2121:
1952:
2069:
1497:
62:
1593:
demand careful thinking about the interpretation of endless sums.
1436:
76:
of this series is −1, which is the limit of the series using the
1387:
similar phenomenon occurs with the divergent geometric series
1834:
Understanding infinity: the mathematics of infinite processes
1320:
may be subtracted from both sides of the equation, yielding
1725:
Gardiner pp. 93–99; the argument on p. 95 for
861:{\displaystyle 1+y+y^{2}+y^{3}+\cdots ={\frac {1}{1-y}}}
928:
The fact that (E) summation assigns a finite value to
179:
since these diverge to infinity, so does the series.
2418:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
2408:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
1818:
1731:
1687:
1563:
1527:
1503:
1479:
1445:
1393:
1361:
1326:
1306:
1283:
1263:
1224:
1197:
1159:
1133:
1096:
1042:
997:
982:
934:
900:
874:
795:
736:
695:
639:
606:
578:
552:
525:
384:
337:
274:
185:
138:
94:
49:, it is characterized by its first term, 1, and its
1781:
p-adic
Numbers, p-adic Analysis, and Zeta-Functions
256:{\displaystyle 2^{0}+2^{1}+\cdots +2^{k}=2^{k+1}-1}
1761:
1708:
1581:
1549:
1512:
1488:
1461:
1423:
1376:
1347:
1312:
1292:
1269:
1245:
1203:
1183:
1145:
1116:
964:
918:
886:
860:
766:
722:
681:
625:
584:
564:
538:
507:
367:
308:
255:
171:
124:
894:These two series are related by the substitution
23:The first four partial sums of 1 + 2 + 4 + 8 + ⋯.
2551:
1939:(November 1983). "Euler and Infinite Series".
1769:is slightly different but has the same spirit.
2054:
2501:Hypergeometric function of a matrix argument
2357:1 + 1/2 + 1/3 + ... (Riemann zeta function)
1879:
1716:are briefly touched on by Hardy p. 19.
1550:{\displaystyle 0.111\ldots ={\frac {1}{9}}}
1300:then it is easily determined. In this case
2061:
2047:
1997:
633:deleted, and it is given by the same rule
375:to the finite value of −1. The associated
309:{\displaystyle \sum _{n=0}^{\infty }2^{n}}
2413:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
1897:
65:, so the sum of this series is infinity.
2068:
1967:
1882:"Euler's 1760 paper on divergent series"
1880:Barbeau, E. J.; Leah, P. J. (May 1976).
1828:
18:
1906:
1778:
682:{\displaystyle f(x)={\frac {1}{1-2x}}.}
323:gives a sum of infinity, including the
2552:
2042:
1935:
1850:
1808:
572:Nonetheless, the so-defined function
1439:appears to have the non-integer sum
2378:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
1615:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
13:
2003:Educational Studies in Mathematics
1872:
1284:
1198:
1140:
786:'s approach to divergent series.)
291:
14:
2586:
2496:Generalized hypergeometric series
172:{\displaystyle 1,3,7,15,\ldots ;}
2534:
2533:
2506:Lauricella hypergeometric series
2224:
626:{\displaystyle x={\frac {1}{2}}}
321:totally regular summation method
2516:Riemann's differential equation
1762:{\displaystyle 1+2+4+8+\cdots }
1582:{\displaystyle 0.999\ldots =1,}
1462:{\displaystyle {\frac {1}{2}}.}
1424:{\displaystyle 1-1+1-1+\cdots }
965:{\displaystyle 1+2+4+8+\cdots }
767:{\displaystyle 1+2+4+8+\cdots }
368:{\displaystyle 1+2+4+8+\cdots }
125:{\displaystyle 1+2+4+8+\cdots }
41:whose terms are the successive
1772:
1719:
1675:
1666:
1657:
1228:
1082:
1052:
705:
699:
649:
643:
539:{\displaystyle {\frac {1}{2}}}
394:
388:
1:
2511:Modular hypergeometric series
2352:1/4 + 1/16 + 1/64 + 1/256 + ⋯
1801:
1246:{\displaystyle z\mapsto 1+2z}
1979:. MAA Online. Archived from
1899:10.1016/0315-0860(76)90030-6
1610:1 − 1 + 2 − 6 + 24 − 120 + ⋯
1513:{\displaystyle 0.999\ldots }
1489:{\displaystyle 0.111\ldots }
83:
7:
2521:Theta hypergeometric series
1814:"De seriebus divergentibus"
1603:
546:so it does not converge at
10:
2591:
2403:Infinite arithmetic series
2347:1/2 + 1/4 + 1/8 + 1/16 + ⋯
2342:1/2 − 1/4 + 1/8 − 1/16 + ⋯
1968:Sandifer, Ed (June 2006).
1153:is a root of the equation
2529:
2486:
2430:
2365:
2334:
2327:
2297:
2266:
2259:
2233:
2222:
2145:
2089:
2080:
1921:10.1080/00033790010028179
1864:QA295 .H29 1967
1836:(Dover ed.). Dover.
1146:{\displaystyle s=\infty }
1650:
1293:{\displaystyle \infty ,}
774:is said to be summable (
723:{\displaystyle f(1)=-1,}
2234:Properties of sequences
1204:{\displaystyle \infty }
1184:{\displaystyle s=1+2s.}
2097:Arithmetic progression
1779:Koblitz, Neal (1984).
1763:
1710:
1709:{\displaystyle s=1+2s}
1583:
1551:
1514:
1490:
1463:
1425:
1378:
1349:
1348:{\displaystyle 0=1+s,}
1314:
1294:
1271:
1247:
1205:
1185:
1147:
1118:
966:
920:
888:
862:
768:
724:
683:
627:
586:
566:
540:
509:
369:
310:
295:
257:
173:
126:
24:
2488:Hypergeometric series
2102:Geometric progression
1764:
1711:
1584:
1552:
1515:
1491:
1464:
1435:), where a series of
1426:
1379:
1377:{\displaystyle s=-1.}
1350:
1315:
1295:
1272:
1248:
1217:Möbius transformation
1206:
1186:
1148:
1119:
967:
921:
919:{\displaystyle y=2x.}
889:
863:
769:
725:
684:
628:
594:analytic continuation
587:
567:
541:
517:radius of convergence
510:
370:
311:
275:
258:
174:
127:
22:
2468:Trigonometric series
2260:Properties of series
2107:Harmonic progression
1941:Mathematics Magazine
1886:Historia Mathematica
1729:
1685:
1561:
1525:
1501:
1477:
1443:
1391:
1359:
1324:
1304:
1281:
1261:
1222:
1195:
1157:
1131:
980:
932:
898:
887:{\displaystyle y=2.}
872:
793:
734:
730:the original series
693:
637:
604:
576:
565:{\displaystyle x=1.}
550:
523:
382:
335:
272:
183:
136:
92:
88:The partial sums of
53:, 2. As a series of
2448:Formal power series
1858:. Clarendon Press.
1589:but the underlying
1127:In a useful sense,
74:Ramanujan summation
2246:Monotonic function
2165:Fibonacci sequence
2015:10.1007/BF00240986
1970:"Divergent series"
1759:
1706:
1579:
1547:
1510:
1486:
1471:recurring decimals
1459:
1421:
1374:
1345:
1310:
1290:
1267:
1243:
1211:is one of the two
1201:
1181:
1143:
1114:
1112:
1109:
1085:
1031:
962:
916:
884:
858:
764:
720:
679:
623:
582:
562:
536:
505:
365:
306:
253:
169:
122:
25:
2560:Binary arithmetic
2547:
2546:
2478:Generating series
2426:
2425:
2398:1 − 2 + 4 − 8 + ⋯
2393:1 + 2 + 4 + 8 + ⋯
2388:1 − 2 + 3 − 4 + ⋯
2383:1 + 2 + 3 + 4 + ⋯
2373:1 + 1 + 1 + 1 + ⋯
2323:
2322:
2251:Periodic sequence
2220:
2219:
2205:Triangular number
2195:Pentagonal number
2175:Heptagonal number
2160:Complete sequence
2082:Integer sequences
1909:Annals of Science
1681:The two roots of
1644:1 + 2 + 4 + ⋯ + 2
1635:1 − 2 + 4 − 8 + ⋯
1630:1 + 2 + 3 + 4 + ⋯
1625:1 − 2 + 3 − 4 + ⋯
1620:1 + 1 + 1 + 1 + ⋯
1545:
1496:and most notably
1454:
1313:{\displaystyle s}
1270:{\displaystyle s}
856:
674:
621:
585:{\displaystyle f}
534:
519:around 0 of only
503:
265:It is written as
70:summation methods
34:1 + 2 + 4 + 8 + ⋯
2582:
2570:Geometric series
2565:Divergent series
2537:
2536:
2463:Dirichlet series
2332:
2331:
2264:
2263:
2228:
2200:Polygonal number
2180:Hexagonal number
2153:
2087:
2086:
2063:
2056:
2049:
2040:
2039:
2034:
1999:Sierpińska, Anna
1994:
1992:
1991:
1985:
1977:How Euler Did It
1974:
1964:
1932:
1903:
1901:
1867:
1856:Divergent Series
1847:
1825:
1795:
1794:
1776:
1770:
1768:
1766:
1765:
1760:
1723:
1717:
1715:
1713:
1712:
1707:
1679:
1673:
1670:
1664:
1663:Hardy p. 10
1661:
1645:
1640:Two's complement
1588:
1586:
1585:
1580:
1556:
1554:
1553:
1548:
1546:
1538:
1519:
1517:
1516:
1511:
1495:
1493:
1492:
1487:
1468:
1466:
1465:
1460:
1455:
1447:
1430:
1428:
1427:
1422:
1383:
1381:
1380:
1375:
1354:
1352:
1351:
1346:
1319:
1317:
1316:
1311:
1299:
1297:
1296:
1291:
1276:
1274:
1273:
1268:
1252:
1250:
1249:
1244:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1152:
1150:
1149:
1144:
1123:
1121:
1120:
1115:
1113:
1089:
1035:
971:
969:
968:
963:
925:
923:
922:
917:
893:
891:
890:
885:
868:and plugging in
867:
865:
864:
859:
857:
855:
841:
830:
829:
817:
816:
782:in reference to
773:
771:
770:
765:
729:
727:
726:
721:
688:
686:
685:
680:
675:
673:
656:
632:
630:
629:
624:
622:
614:
591:
589:
588:
583:
571:
569:
568:
563:
545:
543:
542:
537:
535:
527:
514:
512:
511:
506:
504:
502:
485:
474:
473:
464:
462:
461:
443:
442:
427:
426:
374:
372:
371:
366:
315:
313:
312:
307:
305:
304:
294:
289:
262:
260:
259:
254:
246:
245:
227:
226:
208:
207:
195:
194:
178:
176:
175:
170:
131:
129:
128:
123:
47:geometric series
35:
2590:
2589:
2585:
2584:
2583:
2581:
2580:
2579:
2550:
2549:
2548:
2543:
2525:
2482:
2431:Kinds of series
2422:
2361:
2328:Explicit series
2319:
2293:
2255:
2241:Cauchy sequence
2229:
2216:
2170:Figurate number
2147:
2141:
2132:Powers of three
2076:
2067:
2037:
1989:
1987:
1983:
1972:
1953:10.2307/2690371
1875:
1873:Further reading
1870:
1844:
1810:Euler, Leonhard
1804:
1799:
1798:
1791:
1777:
1773:
1730:
1727:
1726:
1724:
1720:
1686:
1683:
1682:
1680:
1676:
1672:Hardy pp. 8, 10
1671:
1667:
1662:
1658:
1653:
1643:
1606:
1562:
1559:
1558:
1537:
1526:
1523:
1522:
1502:
1499:
1498:
1478:
1475:
1474:
1446:
1444:
1441:
1440:
1433:Grandi's series
1392:
1389:
1388:
1360:
1357:
1356:
1325:
1322:
1321:
1305:
1302:
1301:
1282:
1279:
1278:
1277:; that is, not
1262:
1259:
1258:
1223:
1220:
1219:
1196:
1193:
1192:
1158:
1155:
1154:
1132:
1129:
1128:
1111:
1110:
1094:
1087:
1086:
1040:
1033:
1032:
995:
990:
983:
981:
978:
977:
933:
930:
929:
899:
896:
895:
873:
870:
869:
845:
840:
825:
821:
812:
808:
794:
791:
790:
735:
732:
731:
694:
691:
690:
660:
655:
638:
635:
634:
613:
605:
602:
601:
600:with the point
577:
574:
573:
551:
548:
547:
526:
524:
521:
520:
489:
484:
469:
465:
463:
457:
453:
438:
434:
422:
418:
383:
380:
379:
336:
333:
332:
319:Therefore, any
300:
296:
290:
279:
273:
270:
269:
235:
231:
222:
218:
203:
199:
190:
186:
184:
181:
180:
137:
134:
133:
93:
90:
89:
86:
39:infinite series
33:
17:
16:Infinite series
12:
11:
5:
2588:
2578:
2577:
2575:P-adic numbers
2572:
2567:
2562:
2545:
2544:
2542:
2541:
2530:
2527:
2526:
2524:
2523:
2518:
2513:
2508:
2503:
2498:
2492:
2490:
2484:
2483:
2481:
2480:
2475:
2473:Fourier series
2470:
2465:
2460:
2458:Puiseux series
2455:
2453:Laurent series
2450:
2445:
2440:
2434:
2432:
2428:
2427:
2424:
2423:
2421:
2420:
2415:
2410:
2405:
2400:
2395:
2390:
2385:
2380:
2375:
2369:
2367:
2363:
2362:
2360:
2359:
2354:
2349:
2344:
2338:
2336:
2329:
2325:
2324:
2321:
2320:
2318:
2317:
2312:
2307:
2301:
2299:
2295:
2294:
2292:
2291:
2286:
2281:
2276:
2270:
2268:
2261:
2257:
2256:
2254:
2253:
2248:
2243:
2237:
2235:
2231:
2230:
2223:
2221:
2218:
2217:
2215:
2214:
2213:
2212:
2202:
2197:
2192:
2187:
2182:
2177:
2172:
2167:
2162:
2156:
2154:
2143:
2142:
2140:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2093:
2091:
2084:
2078:
2077:
2066:
2065:
2058:
2051:
2043:
2036:
2035:
2009:(4): 371–396.
1995:
1965:
1947:(5): 307–314.
1933:
1915:(2): 179–199.
1904:
1892:(2): 141–160.
1876:
1874:
1871:
1869:
1868:
1848:
1842:
1826:
1805:
1803:
1800:
1797:
1796:
1789:
1771:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1718:
1705:
1702:
1699:
1696:
1693:
1690:
1674:
1665:
1655:
1654:
1652:
1649:
1648:
1647:
1637:
1632:
1627:
1622:
1617:
1612:
1605:
1602:
1598:2-adic numbers
1578:
1575:
1572:
1569:
1566:
1544:
1541:
1536:
1533:
1530:
1509:
1506:
1485:
1482:
1458:
1453:
1450:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1373:
1370:
1367:
1364:
1344:
1341:
1338:
1335:
1332:
1329:
1309:
1289:
1286:
1266:
1255:Riemann sphere
1242:
1239:
1236:
1233:
1230:
1227:
1200:
1191:(For example,
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1142:
1139:
1136:
1125:
1124:
1108:
1105:
1102:
1099:
1095:
1093:
1090:
1088:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1041:
1039:
1036:
1034:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
996:
994:
991:
989:
986:
985:
961:
958:
955:
952:
949:
946:
943:
940:
937:
915:
912:
909:
906:
903:
883:
880:
877:
854:
851:
848:
844:
839:
836:
833:
828:
824:
820:
815:
811:
807:
804:
801:
798:
784:Leonhard Euler
763:
760:
757:
754:
751:
748:
745:
742:
739:
719:
716:
713:
710:
707:
704:
701:
698:
678:
672:
669:
666:
663:
659:
654:
651:
648:
645:
642:
620:
617:
612:
609:
581:
561:
558:
555:
533:
530:
501:
498:
495:
492:
488:
483:
480:
477:
472:
468:
460:
456:
452:
449:
446:
441:
437:
433:
430:
425:
421:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
364:
361:
358:
355:
352:
349:
346:
343:
340:
317:
316:
303:
299:
293:
288:
285:
282:
278:
252:
249:
244:
241:
238:
234:
230:
225:
221:
217:
214:
211:
206:
202:
198:
193:
189:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
121:
118:
115:
112:
109:
106:
103:
100:
97:
85:
82:
15:
9:
6:
4:
3:
2:
2587:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2557:
2555:
2540:
2532:
2531:
2528:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2493:
2491:
2489:
2485:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2438:Taylor series
2436:
2435:
2433:
2429:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2370:
2368:
2364:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2339:
2337:
2333:
2330:
2326:
2316:
2313:
2311:
2308:
2306:
2303:
2302:
2300:
2296:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2271:
2269:
2265:
2262:
2258:
2252:
2249:
2247:
2244:
2242:
2239:
2238:
2236:
2232:
2227:
2211:
2208:
2207:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2157:
2155:
2151:
2144:
2138:
2135:
2133:
2130:
2128:
2127:Powers of two
2125:
2123:
2120:
2118:
2115:
2113:
2112:Square number
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2094:
2092:
2088:
2085:
2083:
2079:
2075:
2071:
2064:
2059:
2057:
2052:
2050:
2045:
2044:
2041:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1986:on 2013-03-20
1982:
1978:
1971:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1937:Kline, Morris
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1905:
1900:
1895:
1891:
1887:
1883:
1878:
1877:
1865:
1861:
1857:
1853:
1849:
1845:
1843:0-486-42538-X
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1806:
1792:
1790:0-387-96017-1
1786:
1782:
1775:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1722:
1703:
1700:
1697:
1694:
1691:
1688:
1678:
1669:
1660:
1656:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1607:
1601:
1599:
1594:
1592:
1576:
1573:
1570:
1567:
1564:
1542:
1539:
1534:
1531:
1528:
1520:
1507:
1504:
1483:
1480:
1472:
1456:
1451:
1448:
1438:
1434:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1384:
1371:
1368:
1365:
1362:
1342:
1339:
1336:
1333:
1330:
1327:
1307:
1287:
1264:
1256:
1240:
1237:
1234:
1231:
1225:
1218:
1214:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1137:
1134:
1106:
1103:
1100:
1097:
1091:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1049:
1046:
1043:
1037:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
992:
987:
976:
975:
974:
959:
956:
953:
950:
947:
944:
941:
938:
935:
926:
913:
910:
907:
904:
901:
881:
878:
875:
852:
849:
846:
842:
837:
834:
831:
826:
822:
818:
813:
809:
805:
802:
799:
796:
787:
785:
781:
777:
761:
758:
755:
752:
749:
746:
743:
740:
737:
717:
714:
711:
708:
702:
696:
676:
670:
667:
664:
661:
657:
652:
646:
640:
618:
615:
610:
607:
599:
598:complex plane
595:
592:has a unique
579:
559:
556:
553:
531:
528:
518:
499:
496:
493:
490:
486:
481:
478:
475:
470:
466:
458:
454:
450:
447:
444:
439:
435:
431:
428:
423:
419:
415:
412:
409:
406:
403:
400:
397:
391:
385:
378:
362:
359:
356:
353:
350:
347:
344:
341:
338:
330:
326:
322:
301:
297:
286:
283:
280:
276:
268:
267:
266:
263:
250:
247:
242:
239:
236:
232:
228:
223:
219:
215:
212:
209:
204:
200:
196:
191:
187:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
119:
116:
113:
110:
107:
104:
101:
98:
95:
81:
79:
78:2-adic metric
75:
71:
66:
64:
60:
56:
52:
48:
44:
43:powers of two
40:
36:
30:
21:
2443:Power series
2392:
2185:Lucas number
2137:Powers of 10
2117:Cubic number
2006:
2002:
1988:. Retrieved
1981:the original
1976:
1944:
1940:
1912:
1908:
1889:
1885:
1855:
1852:Hardy, G. H.
1833:
1830:Gardiner, A.
1821:
1817:
1780:
1774:
1721:
1677:
1668:
1659:
1595:
1385:
1213:fixed points
1126:
927:
788:
377:power series
318:
264:
87:
67:
55:real numbers
51:common ratio
32:
26:
2310:Conditional
2298:Convergence
2289:Telescoping
2274:Alternating
2190:Pell number
780:G. H. Hardy
29:mathematics
2554:Categories
2335:Convergent
2279:Convergent
1990:2007-02-17
1824:: 205–237.
1802:References
325:Cesàro sum
2366:Divergent
2284:Divergent
2146:Advanced
2122:Factorial
2070:Sequences
2031:144880659
1929:143992318
1832:(2002) .
1757:⋯
1568:…
1532:…
1508:…
1484:…
1419:⋯
1410:−
1398:−
1369:−
1285:∞
1229:↦
1199:∞
1141:∞
1080:⋯
1029:⋯
960:⋯
850:−
835:⋯
762:⋯
712:−
665:−
494:−
479:⋯
448:⋯
363:⋯
292:∞
277:∑
248:−
213:⋯
164:…
120:⋯
84:Summation
2539:Category
2305:Absolute
1854:(1949).
1812:(1760).
1604:See also
1437:integers
329:Abel sum
63:infinity
59:diverges
2315:Uniform
2023:3482354
1961:2690371
1253:on the
1215:of the
596:to the
45:. As a
37:is the
2267:Series
2074:series
2029:
2021:
1959:
1927:
1862:
1840:
1787:
1591:proofs
689:Since
515:has a
2210:array
2090:Basic
2027:S2CID
2019:JSTOR
1984:(PDF)
1973:(PDF)
1957:JSTOR
1925:S2CID
1651:Notes
1565:0.999
1529:0.111
1505:0.999
1481:0.111
2150:list
2072:and
1838:ISBN
1785:ISBN
1557:and
327:and
132:are
2011:doi
1949:doi
1917:doi
1894:doi
1860:LCC
1473:as
1355:so
61:to
57:it
27:In
2556::
2025:.
2017:.
2007:18
2005:.
1975:.
1955:.
1945:56
1943:.
1923:.
1913:59
1911:.
1888:.
1884:.
1820:.
1816:.
1372:1.
1023:16
882:2.
560:1.
158:15
80:.
31:,
2152:)
2148:(
2062:e
2055:t
2048:v
2033:.
2013::
1993:.
1963:.
1951::
1931:.
1919::
1902:.
1896::
1890:3
1866:.
1846:.
1822:5
1793:.
1754:+
1751:8
1748:+
1745:4
1742:+
1739:2
1736:+
1733:1
1704:s
1701:2
1698:+
1695:1
1692:=
1689:s
1646:.
1577:,
1574:1
1571:=
1543:9
1540:1
1535:=
1457:.
1452:2
1449:1
1431:(
1416:+
1413:1
1407:1
1404:+
1401:1
1395:1
1366:=
1363:s
1343:,
1340:s
1337:+
1334:1
1331:=
1328:0
1308:s
1288:,
1265:s
1241:z
1238:2
1235:+
1232:1
1226:z
1179:.
1176:s
1173:2
1170:+
1167:1
1164:=
1161:s
1138:=
1135:s
1107:s
1104:2
1101:+
1098:1
1092:=
1083:)
1077:+
1074:8
1071:+
1068:4
1065:+
1062:2
1059:+
1056:1
1053:(
1050:2
1047:+
1044:1
1038:=
1026:+
1020:+
1017:8
1014:+
1011:4
1008:+
1005:2
1002:+
999:1
993:=
988:s
957:+
954:8
951:+
948:4
945:+
942:2
939:+
936:1
914:.
911:x
908:2
905:=
902:y
879:=
876:y
853:y
847:1
843:1
838:=
832:+
827:3
823:y
819:+
814:2
810:y
806:+
803:y
800:+
797:1
776:E
759:+
756:8
753:+
750:4
747:+
744:2
741:+
738:1
718:,
715:1
709:=
706:)
703:1
700:(
697:f
677:.
671:x
668:2
662:1
658:1
653:=
650:)
647:x
644:(
641:f
619:2
616:1
611:=
608:x
580:f
557:=
554:x
532:2
529:1
500:x
497:2
491:1
487:1
482:=
476:+
471:n
467:x
459:n
455:2
451:+
445:+
440:3
436:x
432:8
429:+
424:2
420:x
416:4
413:+
410:x
407:2
404:+
401:1
398:=
395:)
392:x
389:(
386:f
360:+
357:8
354:+
351:4
348:+
345:2
342:+
339:1
302:n
298:2
287:0
284:=
281:n
251:1
243:1
240:+
237:k
233:2
229:=
224:k
220:2
216:+
210:+
205:1
201:2
197:+
192:0
188:2
167:;
161:,
155:,
152:7
149:,
146:3
143:,
140:1
117:+
114:8
111:+
108:4
105:+
102:2
99:+
96:1
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