2237:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 16.
2175:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 16.
728:
1939:
2107:
890:
If the original series, on the left hand side, is only slowly converging, the forward differences will tend to become small quite rapidly; the additional power of two further improves the rate at which the right hand side converges.
885:
363:
251:
1079:
2451:
Brezinski Claude, Redivo-Zaglia
Michela and Saad Yousef : "Shanks Sequence Transformations and Anderson Acceleration", SIAM Review, Vol.60, No.3 (2018), pp.646–669. doi:10.1137/17M1120725 .
183:
122:
1803:
961:
575:
1758:
1410:
1620:
1556:
1448:
1327:
1284:
1249:
531:
501:
2440:
1492:
751:
1351:
401:
278:
1718:
1669:
1585:
1521:
1111:
993:
1784:
1211:
1161:
1689:
1640:
1181:
551:
1950:
1849:
763:
286:
1786:
in that series expansion will thus yield a series such that if it converges, it will converge to the same value as the original series.
2461:
Brezinski Claude and Redivo-Zaglia
Michela : "Extrapolation and Rational Approximation", Springer, ISBN 978-3-030-58417-7 (2020).
2422:
Brezinski Claude and Redivo-Zaglia
Michela : "The genesis and early developments of Aitken's process, Shanks transformation, the
2242:
2180:
569:, offering improved convergence, is Euler's transform. It is intended to be applied to an alternating series; it is given by
2516:
1008:
427:
423:
194:
1114:
430:. A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including
136:
2471:
2322:
1838:
723:{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(\Delta ^{n}a)_{0}}{2^{n+1}}}}
73:
2134:
912:
2139:
895:
566:
2476:
2281:
2511:
2506:
2501:
443:
2496:
754:
2448:
Sidi Avram : "Vector
Extrapolation Methods with Applications", SIAM, ISBN 978-1-61197-495-9 (2017).
2455:
1359:
459:
2445:
Delahaye J. P. : "Sequence
Transformations", Springer-Verlag, Berlin, ISBN 978-3540152835 (1988).
2289:
1723:
44:. Series acceleration techniques may also be used, for example, to obtain a variety of identities on
2442:-algorithm, and related fixed point methods", Numerical Algorithms, Vol.80, No.1, (2019), pp.11-133.
431:
1126:
411:
373:
25:
1843:
A simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method,
1809:
Especially nonlinear sequence transformations often provide powerful numerical methods for the
1590:
1526:
1418:
1297:
1254:
1219:
506:
476:
414:), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.
2425:
1453:
996:
2333:
2232:
2170:
2129:
1799:
1134:
1130:
736:
53:
41:
2355:
1183:
will converge very slowly. One can then improve the convergence of the series by means of a
2394:
2260:
2198:
1826:
1336:
435:
386:
377:
263:
33:
1694:
1645:
1561:
1497:
1087:
969:
8:
2113:
1822:
1763:
1190:
1140:
128:
29:
2398:
1795:
2412:
2384:
2102:{\displaystyle s'_{n}=s_{n+2}-{\frac {(s_{n+2}-s_{n+1})^{2}}{s_{n+2}-2s_{n+1}+s_{n}}}.}
1674:
1625:
1166:
536:
470:
455:
37:
2406:
2116:
of a slowly converging sequence; heuristically, it eliminates the largest part of the
2352:
2318:
2264:
2248:
2238:
2220:
2202:
2186:
2176:
2158:
1818:
1184:
45:
2373:, Journal of Computational and Applied Mathematics, vol. 122, no. 1–2, p 81 (2000).
2402:
1934:{\displaystyle \mathbb {A} :S\to S'=\mathbb {A} (S)={(s'_{n})}_{n\in \mathbb {N} }}
1814:
447:
369:
2256:
2234:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
2194:
2172:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
49:
2228:
2166:
894:
A particularly efficient numerical implementation of the Euler transform is the
2117:
880:{\displaystyle (\Delta ^{n}a)_{0}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{n-k}.}
407:
2490:
1118:
451:
2224:
2162:
1122:
439:
1163:
is close to or on the boundary of the disk of convergence, the series for
358:{\displaystyle \lim _{n\to \infty }{\frac {s'_{n}-\ell }{s_{n}-\ell }}=0.}
17:
2375:
Homeier, H. H. H. (2000). "Scalar Levin-type sequence transformations".
462:
in 1956; the Levin u-transform; and the Wilf-Zeilberger-Ekhad method or
56:
gives some of the classic, well-known hypergeometric series identities.
2285:
1287:
2416:
2389:
2360:
1810:
463:
381:
2268:
2206:
1187:
that moves the singularities such that the point that is mapped to
65:
2350:
2252:
2190:
473:, several powerful techniques, offering convergence rates from
406:
The mappings from the original to the transformed series may be
2481:
36:. Techniques for series acceleration are often applied in
1794:
Examples of such nonlinear sequence transformations are
1286:, and one usually chooses a function that has a finite
1789:
1329:
without loss of generality, as one can always rescale
2428:
1953:
1852:
1766:
1726:
1697:
1677:
1648:
1628:
1593:
1564:
1529:
1500:
1456:
1421:
1362:
1339:
1300:
1257:
1222:
1193:
1169:
1143:
1090:
1074:{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}.}
1011:
972:
915:
766:
739:
578:
539:
509:
479:
438:
in the early 20th century but also known and used by
422:
Two classical techniques for series acceleration are
389:
289:
266:
197:
139:
76:
2112:
This transformation is commonly used to improve the
2434:
2101:
1933:
1778:
1752:
1712:
1683:
1663:
1634:
1614:
1579:
1550:
1515:
1486:
1442:
1404:
1345:
1321:
1278:
1243:
1205:
1175:
1155:
1105:
1073:
987:
955:
879:
745:
722:
545:
525:
495:
395:
357:
272:
245:
177:
116:
852:
839:
246:{\displaystyle S'=\{s'_{n}\}_{n\in \mathbb {N} }}
2488:
2377:Journal of Computational and Applied Mathematics
2219:
2157:
291:
178:{\displaystyle \lim _{n\to \infty }s_{n}=\ell ,}
141:
2458:", Numerical Algorithms, Vol.80(2019), pp.5-10.
1213:ends up deeper in the new disk of convergence.
117:{\displaystyle S=\{s_{n}\}_{n\in \mathbb {N} }}
2290:Convergence Acceleration of Alternating Series
280:than the original sequence, in the sense that
40:, where they are used to improve the speed of
956:{\displaystyle S=\sum _{n=0}^{\infty }a_{n}}
226:
209:
97:
83:
2477:GNU Scientific Library, Series Acceleration
188:an accelerated series is a second sequence
2454:Brezinski Claude : "Reminiscences of
2371:Scalar Levin-Type Sequence Transformations
2338:Extrapolation Methods. Theory and Practice
2482:Digital Library of Mathematical Functions
2388:
1925:
1879:
1854:
237:
108:
1494:. We can obtain the series expansion of
2374:
1825:, and may be used as highly effective
2489:
2343:G. A. Baker Jr. and P. Graves-Morris,
2351:
2317:, (1987) Cambridge University Press,
901:
560:
2284:, Fernando Rodriguez Villegas, and
1804:Levin-type sequence transformations
1790:Non-linear sequence transformations
450:in 1926 but also known and used by
13:
2472:Convergence acceleration of series
1728:
1691:terms of the series expansion for
1642:terms of the series expansion for
1594:
1536:
1422:
1384:
1340:
1301:
1258:
1229:
1043:
938:
843:
771:
740:
679:
648:
595:
301:
151:
14:
2528:
2465:
1405:{\displaystyle g(w)=f(\Phi (w)).}
428:Kummer's transformation of series
2135:Minimum polynomial extrapolation
1832:
1353:. We then consider the function
757:, for which one has the formula
424:Euler's transformation of series
1753:{\displaystyle \Phi '(0)\neq 0}
2303:
2275:
2213:
2151:
2140:Van Wijngaarden transformation
2031:
1992:
1913:
1897:
1889:
1883:
1864:
1839:Aitken's delta-squared process
1741:
1735:
1707:
1701:
1658:
1652:
1603:
1597:
1574:
1568:
1545:
1539:
1510:
1504:
1481:
1475:
1466:
1460:
1431:
1425:
1396:
1393:
1387:
1381:
1372:
1366:
1310:
1304:
1267:
1261:
1238:
1232:
1100:
1094:
1021:
1015:
982:
976:
896:van Wijngaarden transformation
827:
817:
784:
767:
692:
675:
663:
653:
610:
600:
567:linear sequence transformation
553:terms, are described by Cohen
298:
148:
1:
2407:10.1016/S0377-0427(00)00359-9
2145:
1251:needs to be chosen such that
59:
1137:of the series. If the point
444:Aitken delta-squared process
368:If the original sequence is
7:
2517:Series acceleration methods
2123:
1821:that arise for instance in
1558:in the series expansion of
755:forward difference operator
417:
410:(as defined in the article
10:
2533:
1836:
1615:{\displaystyle \Phi (0)=0}
1551:{\displaystyle z=\Phi (w)}
1443:{\displaystyle \Phi (1)=1}
1322:{\displaystyle \Phi (1)=1}
1279:{\displaystyle \Phi (0)=0}
1244:{\displaystyle z=\Phi (w)}
526:{\displaystyle 17.93^{-n}}
496:{\displaystyle 5.828^{-n}}
24:is one of a collection of
2435:{\displaystyle \epsilon }
2356:"Convergence Improvement"
1487:{\displaystyle f(1)=g(1)}
1294:= 0. One can assume that
454:in the 18th century; the
2294:Experimental Mathematics
1216:The conformal transform
432:Richardson extrapolation
412:sequence transformations
26:sequence transformations
2369:Herbert H. H. Homeier:
2347:, Cambridge U.P., 1996.
2229:"Chapter 3, eqn 3.6.26"
2167:"Chapter 3, eqn 3.6.27"
1131:essential singularities
746:{\displaystyle \Delta }
374:sequence transformation
2436:
2340:, North-Holland, 1991.
2315:Numerical Recipes in C
2103:
1935:
1780:
1754:
1714:
1685:
1665:
1636:
1616:
1581:
1552:
1517:
1488:
1444:
1406:
1347:
1323:
1280:
1245:
1207:
1177:
1157:
1107:
1075:
1047:
989:
957:
942:
881:
816:
747:
724:
652:
599:
547:
527:
497:
397:
359:
274:
247:
179:
118:
2437:
2130:Shanks transformation
2104:
1936:
1827:extrapolation methods
1800:Shanks transformation
1781:
1755:
1715:
1686:
1671:will yield the first
1666:
1637:
1617:
1582:
1553:
1518:
1489:
1445:
1407:
1348:
1346:{\displaystyle \Phi }
1324:
1281:
1246:
1208:
1178:
1158:
1135:radius of convergence
1108:
1076:
1027:
990:
958:
922:
882:
796:
748:
725:
632:
579:
565:A basic example of a
548:
528:
498:
398:
396:{\displaystyle \ell }
360:
275:
273:{\displaystyle \ell }
248:
180:
119:
54:hypergeometric series
42:numerical integration
2426:
1951:
1850:
1764:
1724:
1713:{\displaystyle g(w)}
1695:
1675:
1664:{\displaystyle f(z)}
1646:
1626:
1591:
1580:{\displaystyle f(z)}
1562:
1527:
1516:{\displaystyle g(w)}
1498:
1454:
1419:
1360:
1337:
1298:
1255:
1220:
1191:
1167:
1141:
1106:{\displaystyle f(z)}
1088:
1009:
988:{\displaystyle f(1)}
970:
913:
764:
737:
576:
537:
507:
477:
436:Lewis Fry Richardson
387:
378:extrapolation method
287:
264:
195:
137:
74:
2512:Perturbation theory
2507:Summability methods
2502:Asymptotic analysis
2399:2000JCoAM.122...81H
2114:rate of convergence
1966:
1912:
1823:perturbation theory
1779:{\displaystyle w=1}
1206:{\displaystyle z=1}
1156:{\displaystyle z=1}
1133:), which limit the
533:for a summation of
321:
224:
30:rate of convergence
22:series acceleration
2497:Numerical analysis
2432:
2353:Weisstein, Eric W.
2345:Padé Approximants
2325:(See section 5.1).
2309:William H. Press,
2221:Abramowitz, Milton
2159:Abramowitz, Milton
2099:
1954:
1931:
1900:
1776:
1750:
1710:
1681:
1661:
1632:
1612:
1577:
1548:
1513:
1484:
1440:
1402:
1343:
1319:
1276:
1241:
1203:
1173:
1153:
1103:
1071:
985:
966:can be written as
953:
902:Conformal mappings
877:
743:
720:
543:
523:
493:
471:alternating series
393:
355:
309:
305:
270:
243:
212:
175:
155:
114:
38:numerical analysis
28:for improving the
2332:C. Brezinski and
2300::1 (2000) page 3.
2244:978-0-486-61272-0
2225:Stegun, Irene Ann
2182:978-0-486-61272-0
2163:Stegun, Irene Ann
2094:
1819:asymptotic series
1796:Padé approximants
1684:{\displaystyle n}
1635:{\displaystyle n}
1185:conformal mapping
1176:{\displaystyle S}
850:
718:
561:Euler's transform
546:{\displaystyle n}
347:
290:
140:
46:special functions
2524:
2441:
2439:
2438:
2433:
2410:
2392:
2366:
2365:
2334:M. Redivo Zaglia
2326:
2307:
2301:
2279:
2273:
2272:
2227:, eds. (1983) .
2217:
2211:
2210:
2165:, eds. (1983) .
2155:
2108:
2106:
2105:
2100:
2095:
2093:
2092:
2091:
2079:
2078:
2057:
2056:
2040:
2039:
2038:
2029:
2028:
2010:
2009:
1990:
1985:
1984:
1962:
1940:
1938:
1937:
1932:
1930:
1929:
1928:
1916:
1908:
1882:
1874:
1857:
1815:divergent series
1785:
1783:
1782:
1777:
1759:
1757:
1756:
1751:
1734:
1719:
1717:
1716:
1711:
1690:
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1687:
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1670:
1668:
1667:
1662:
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1621:
1619:
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1613:
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1584:
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1557:
1555:
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1549:
1522:
1520:
1519:
1514:
1493:
1491:
1490:
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1441:
1411:
1409:
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1403:
1352:
1350:
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1328:
1326:
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1320:
1285:
1283:
1282:
1277:
1250:
1248:
1247:
1242:
1212:
1210:
1209:
1204:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1112:
1110:
1109:
1104:
1080:
1078:
1077:
1072:
1067:
1066:
1057:
1056:
1046:
1041:
994:
992:
991:
986:
962:
960:
959:
954:
952:
951:
941:
936:
886:
884:
883:
878:
873:
872:
857:
856:
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842:
835:
834:
815:
810:
792:
791:
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778:
752:
750:
749:
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729:
727:
726:
721:
719:
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687:
686:
673:
671:
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651:
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628:
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552:
550:
549:
544:
532:
530:
529:
524:
522:
521:
502:
500:
499:
494:
492:
491:
448:Alexander Aitken
446:, introduced by
434:, introduced by
402:
400:
399:
394:
364:
362:
361:
356:
348:
346:
339:
338:
328:
317:
307:
304:
279:
277:
276:
271:
258:converges faster
252:
250:
249:
244:
242:
241:
240:
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205:
184:
182:
181:
176:
165:
164:
154:
123:
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120:
115:
113:
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111:
95:
94:
2532:
2531:
2527:
2526:
2525:
2523:
2522:
2521:
2487:
2486:
2468:
2427:
2424:
2423:
2383:(1–2): 81–147.
2329:
2308:
2304:
2280:
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2245:
2218:
2214:
2183:
2156:
2152:
2148:
2126:
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2018:
2014:
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1995:
1991:
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1643:
1627:
1624:
1623:
1592:
1589:
1588:
1563:
1560:
1559:
1528:
1525:
1524:
1499:
1496:
1495:
1455:
1452:
1451:
1420:
1417:
1416:
1361:
1358:
1357:
1338:
1335:
1334:
1299:
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1295:
1256:
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1221:
1218:
1217:
1192:
1189:
1188:
1168:
1165:
1164:
1142:
1139:
1138:
1125:singularities,
1089:
1086:
1085:
1062:
1058:
1052:
1048:
1042:
1031:
1010:
1007:
1006:
971:
968:
967:
947:
943:
937:
926:
914:
911:
910:
904:
862:
858:
851:
838:
837:
836:
830:
826:
811:
800:
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734:
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702:
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691:
682:
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666:
662:
647:
636:
623:
619:
613:
609:
594:
583:
577:
574:
573:
563:
538:
535:
534:
514:
510:
508:
505:
504:
503:all the way to
484:
480:
478:
475:
474:
440:Katahiro Takebe
420:
388:
385:
384:
334:
330:
329:
313:
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144:
138:
135:
134:
107:
100:
96:
90:
86:
75:
72:
71:
62:
52:applied to the
50:Euler transform
12:
11:
5:
2530:
2520:
2519:
2514:
2509:
2504:
2499:
2485:
2484:
2479:
2474:
2467:
2466:External links
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2212:
2181:
2149:
2147:
2144:
2143:
2142:
2137:
2132:
2125:
2122:
2118:absolute error
2110:
2109:
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1973:
1969:
1965:
1961:
1957:
1942:
1941:
1927:
1923:
1920:
1915:
1911:
1907:
1903:
1899:
1894:
1891:
1888:
1885:
1881:
1877:
1873:
1870:
1866:
1863:
1860:
1856:
1837:Main article:
1834:
1831:
1791:
1788:
1775:
1772:
1769:
1749:
1746:
1743:
1740:
1737:
1733:
1730:
1709:
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1703:
1700:
1680:
1660:
1657:
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1631:
1611:
1608:
1605:
1602:
1599:
1596:
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1573:
1570:
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1547:
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1541:
1538:
1535:
1532:
1512:
1509:
1506:
1503:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1439:
1436:
1433:
1430:
1427:
1424:
1413:
1412:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1342:
1318:
1315:
1312:
1309:
1306:
1303:
1275:
1272:
1269:
1266:
1263:
1260:
1240:
1237:
1234:
1231:
1228:
1225:
1202:
1199:
1196:
1172:
1152:
1149:
1146:
1102:
1099:
1096:
1093:
1082:
1081:
1070:
1065:
1061:
1055:
1051:
1045:
1040:
1037:
1034:
1030:
1026:
1023:
1020:
1017:
1014:
1002:is defined as
984:
981:
978:
975:
964:
963:
950:
946:
940:
935:
932:
929:
925:
921:
918:
903:
900:
888:
887:
876:
871:
868:
865:
861:
854:
849:
846:
841:
833:
829:
825:
822:
819:
814:
809:
806:
803:
799:
795:
790:
786:
782:
777:
773:
769:
742:
731:
730:
715:
712:
709:
705:
698:
694:
690:
685:
681:
677:
669:
665:
661:
658:
655:
650:
645:
642:
639:
635:
631:
626:
622:
616:
612:
608:
605:
602:
597:
592:
589:
586:
582:
562:
559:
542:
520:
517:
513:
490:
487:
483:
456:epsilon method
419:
416:
392:
366:
365:
354:
351:
345:
342:
337:
333:
327:
324:
320:
316:
312:
303:
300:
297:
293:
269:
254:
253:
239:
235:
232:
228:
223:
219:
215:
211:
208:
204:
201:
186:
185:
174:
171:
168:
163:
159:
153:
150:
147:
143:
125:
124:
110:
106:
103:
99:
93:
89:
85:
82:
79:
61:
58:
9:
6:
4:
3:
2:
2529:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2494:
2492:
2483:
2480:
2478:
2475:
2473:
2470:
2469:
2460:
2457:
2453:
2450:
2447:
2444:
2429:
2421:
2418:
2414:
2408:
2404:
2400:
2396:
2391:
2386:
2382:
2378:
2372:
2368:
2363:
2362:
2357:
2354:
2349:
2346:
2342:
2339:
2335:
2331:
2330:
2324:
2323:0-521-43108-5
2320:
2316:
2312:
2306:
2299:
2295:
2291:
2287:
2283:
2278:
2270:
2266:
2262:
2258:
2254:
2250:
2246:
2240:
2236:
2235:
2230:
2226:
2222:
2216:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2178:
2174:
2173:
2168:
2164:
2160:
2154:
2150:
2141:
2138:
2136:
2133:
2131:
2128:
2127:
2121:
2119:
2115:
2096:
2088:
2084:
2080:
2075:
2072:
2069:
2065:
2061:
2058:
2053:
2050:
2047:
2043:
2035:
2025:
2022:
2019:
2015:
2011:
2006:
2003:
2000:
1996:
1986:
1981:
1978:
1975:
1971:
1967:
1963:
1959:
1955:
1947:
1946:
1945:
1921:
1918:
1909:
1905:
1901:
1892:
1886:
1875:
1871:
1868:
1861:
1858:
1846:
1845:
1844:
1840:
1833:Aitken method
1830:
1828:
1824:
1820:
1816:
1812:
1807:
1805:
1801:
1797:
1787:
1773:
1770:
1767:
1747:
1744:
1738:
1731:
1704:
1698:
1678:
1655:
1649:
1629:
1609:
1606:
1600:
1571:
1565:
1542:
1533:
1530:
1507:
1501:
1478:
1472:
1469:
1463:
1457:
1437:
1434:
1428:
1399:
1390:
1378:
1375:
1369:
1363:
1356:
1355:
1354:
1332:
1316:
1313:
1307:
1293:
1289:
1273:
1270:
1264:
1235:
1226:
1223:
1214:
1200:
1197:
1194:
1186:
1170:
1150:
1147:
1144:
1136:
1132:
1128:
1124:
1120:
1119:complex plane
1116:
1115:singularities
1097:
1091:
1084:The function
1068:
1063:
1059:
1053:
1049:
1038:
1035:
1032:
1028:
1024:
1018:
1012:
1005:
1004:
1003:
1001:
998:
979:
973:
948:
944:
933:
930:
927:
923:
919:
916:
909:
908:
907:
899:
897:
892:
874:
869:
866:
863:
859:
847:
844:
831:
823:
820:
812:
807:
804:
801:
797:
793:
788:
780:
775:
760:
759:
758:
756:
713:
710:
707:
703:
696:
688:
683:
667:
659:
656:
643:
640:
637:
633:
629:
624:
620:
614:
606:
603:
590:
587:
584:
580:
572:
571:
570:
568:
558:
556:
540:
518:
515:
511:
488:
485:
481:
472:
467:
465:
461:
457:
453:
452:Takakazu Seki
449:
445:
442:in 1722; the
441:
437:
433:
429:
425:
415:
413:
409:
404:
390:
383:
379:
375:
371:
352:
349:
343:
340:
335:
331:
325:
322:
318:
314:
310:
295:
283:
282:
281:
267:
259:
233:
230:
221:
217:
213:
206:
202:
199:
191:
190:
189:
172:
169:
166:
161:
157:
145:
133:
132:
131:
130:
104:
101:
91:
87:
80:
77:
70:
69:
68:
67:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
2417:math/0005209
2390:math/0005209
2380:
2376:
2370:
2359:
2344:
2337:
2314:
2310:
2305:
2297:
2293:
2277:
2233:
2215:
2171:
2153:
2111:
1943:
1842:
1808:
1793:
1622:; the first
1414:
1333:to redefine
1330:
1291:
1215:
1123:branch point
1083:
999:
995:, where the
965:
905:
893:
889:
732:
564:
554:
468:
421:
405:
367:
257:
255:
187:
126:
63:
48:. Thus, the
21:
15:
2282:Henri Cohen
1944:defined by
1523:by putting
376:acts as an
18:mathematics
2491:Categories
2456:Peter Wynn
2286:Don Zagier
2146:References
1760:. Putting
1450:, we have
1288:derivative
460:Peter Wynn
60:Definition
2430:ϵ
2361:MathWorld
2059:−
2012:−
1987:−
1922:∈
1865:→
1811:summation
1745:≠
1729:Φ
1595:Φ
1537:Φ
1423:Φ
1385:Φ
1341:Φ
1302:Φ
1259:Φ
1230:Φ
1113:can have
1044:∞
1029:∑
939:∞
924:∑
906:A series
867:−
821:−
798:∑
772:Δ
741:Δ
680:Δ
657:−
649:∞
634:∑
604:−
596:∞
581:∑
516:−
486:−
464:WZ method
458:given by
391:ℓ
382:antilimit
370:divergent
344:ℓ
341:−
326:ℓ
323:−
302:∞
299:→
268:ℓ
234:∈
170:ℓ
152:∞
149:→
127:having a
105:∈
2269:65-12253
2253:64-60036
2207:65-12253
2191:64-60036
2124:See also
1964:′
1910:′
1872:′
1732:′
1587:because
997:function
418:Overview
319:′
222:′
203:′
66:sequence
64:Given a
2395:Bibcode
2261:0167642
2199:0167642
1117:in the
753:is the
380:to the
2321:
2311:et al.
2267:
2259:
2251:
2241:
2205:
2197:
2189:
2179:
1802:, and
1798:, the
1415:Since
733:where
408:linear
372:, the
256:which
34:series
2413:arXiv
2385:arXiv
1127:poles
555:et al
512:17.93
482:5.828
129:limit
32:of a
2319:ISBN
2265:LCCN
2249:LCCN
2239:ISBN
2203:LCCN
2187:LCCN
2177:ISBN
469:For
426:and
2403:doi
2381:122
2292:",
2288:, "
1817:or
1813:of
1720:if
1290:at
1129:or
292:lim
260:to
142:lim
16:In
2493::
2411:,
2401:.
2393:.
2379:.
2358:.
2336:,
2313:,
2296:,
2263:.
2257:MR
2255:.
2247:.
2231:.
2223:;
2201:.
2195:MR
2193:.
2185:.
2169:.
2161:;
2120:.
1829:.
1806:.
898:.
557:.
466:.
403:.
353:0.
20:,
2419:.
2415::
2409:.
2405::
2397::
2387::
2364:.
2298:9
2271:.
2209:.
2097:.
2089:n
2085:s
2081:+
2076:1
2073:+
2070:n
2066:s
2062:2
2054:2
2051:+
2048:n
2044:s
2036:2
2032:)
2026:1
2023:+
2020:n
2016:s
2007:2
2004:+
2001:n
1997:s
1993:(
1982:2
1979:+
1976:n
1972:s
1968:=
1960:n
1956:s
1926:N
1919:n
1914:)
1906:n
1902:s
1898:(
1893:=
1890:)
1887:S
1884:(
1880:A
1876:=
1869:S
1862:S
1859::
1855:A
1774:1
1771:=
1768:w
1748:0
1742:)
1739:0
1736:(
1708:)
1705:w
1702:(
1699:g
1679:n
1659:)
1656:z
1653:(
1650:f
1630:n
1610:0
1607:=
1604:)
1601:0
1598:(
1575:)
1572:z
1569:(
1566:f
1546:)
1543:w
1540:(
1534:=
1531:z
1511:)
1508:w
1505:(
1502:g
1482:)
1479:1
1476:(
1473:g
1470:=
1467:)
1464:1
1461:(
1458:f
1438:1
1435:=
1432:)
1429:1
1426:(
1400:.
1397:)
1394:)
1391:w
1388:(
1382:(
1379:f
1376:=
1373:)
1370:w
1367:(
1364:g
1331:w
1317:1
1314:=
1311:)
1308:1
1305:(
1292:w
1274:0
1271:=
1268:)
1265:0
1262:(
1239:)
1236:w
1233:(
1227:=
1224:z
1201:1
1198:=
1195:z
1171:S
1151:1
1148:=
1145:z
1121:(
1101:)
1098:z
1095:(
1092:f
1069:.
1064:n
1060:z
1054:n
1050:a
1039:0
1036:=
1033:n
1025:=
1022:)
1019:z
1016:(
1013:f
1000:f
983:)
980:1
977:(
974:f
949:n
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934:0
931:=
928:n
920:=
917:S
875:.
870:k
864:n
860:a
853:)
848:k
845:n
840:(
832:k
828:)
824:1
818:(
813:n
808:0
805:=
802:k
794:=
789:0
785:)
781:a
776:n
768:(
714:1
711:+
708:n
704:2
697:0
693:)
689:a
684:n
676:(
668:n
664:)
660:1
654:(
644:0
641:=
638:n
630:=
625:n
621:a
615:n
611:)
607:1
601:(
591:0
588:=
585:n
541:n
519:n
489:n
350:=
336:n
332:s
315:n
311:s
296:n
238:N
231:n
227:}
218:n
214:s
210:{
207:=
200:S
173:,
167:=
162:n
158:s
146:n
109:N
102:n
98:}
92:n
88:s
84:{
81:=
78:S
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