27:
1487:
350:
1228:
1217:
864:
630:
482:
157:
1046:
1482:{\displaystyle \sum _{j=0}^{\infty }z^{j}=\sum _{i=0}^{\infty }{\frac {1}{(1+y)^{i+1}}}\sum _{j=0}^{i}{\binom {i}{j}}y^{j+1}z^{j}={\frac {y}{1+y}}\sum _{i=0}^{\infty }\left({\frac {1+yz}{1+y}}\right)^{i}}
719:
1025:
724:
952:
95:
is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ
493:
368:
112:
of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.
345:{\displaystyle _{E_{y}}\,\sum _{j=0}^{\infty }a_{j}:=\sum _{i=0}^{\infty }{\frac {1}{(1+y)^{i+1}}}\sum _{j=0}^{i}{\binom {i}{j}}y^{j+1}a_{j}.}
648:
48:
961:
1652:
1633:
1614:
70:
1584:
1569:
355:
If all the formal sums actually converge, the Euler sum will equal the left hand side. However, using Euler summation can
896:
1677:
1672:
1212:{\displaystyle {\frac {1}{1-2^{k+1}}}\sum _{i=0}^{k}{\frac {1}{2^{i+1}}}\sum _{j=0}^{i}{\binom {i}{j}}(-1)^{j}(j+1)^{k}}
1579:
41:
35:
362:
To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because
359:(this is especially useful for alternating series); sometimes it can also give a useful meaning to divergent sums.
1589:
52:
1574:
1220:
1036:
874:
1040:
1028:
1564:
1554:
356:
859:{\displaystyle \sum _{i=0}^{k}{\frac {1}{2^{i+1}}}\sum _{j=0}^{i}{\binom {i}{j}}(-1)^{j}P_{k}(j),}
1604:
1544:
625:{\displaystyle _{E_{y_{1}}}{}_{E_{y_{2}}}\sum =\,_{E_{\frac {y_{1}y_{2}}{1+y_{1}+y_{2}}}}\sum .}
1648:
1629:
1610:
1559:
955:
84:
123:≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than
88:
1549:
1035:
is positive, but applying Euler summation to the zeta function (or rather, to the related
124:
105:
132:
477:{\displaystyle y^{j+1}\sum _{i=j}^{\infty }{\binom {i}{j}}{\frac {1}{(1+y)^{i+1}}}=1.}
1666:
890:, so in this case Euler summation reduces an infinite series to a finite sum.
115:
Euler summation can be generalized into a family of methods denoted (E,
487:
This method itself cannot be improved by iterated application, as
147:
we may define the Euler sum (if it converges for that value of
20:
1626:
Borel's
Methods of Summability: Theory and Applications
1031:). Indeed, the formal sum in this case diverges since
1231:
1049:
964:
899:
727:
714:{\displaystyle \sum _{j=0}^{\infty }(-1)^{j}P_{k}(j)}
651:
496:
371:
160:
151:) corresponding to a particular formal summation as:
1481:
1211:
1020:{\displaystyle {\frac {B_{k+1}}{k+1}}=-\zeta (-k)}
1019:
946:
858:
713:
624:
476:
344:
1361:
1348:
1162:
1149:
809:
796:
425:
412:
307:
294:
108:converges to a sum, then that sum is called the
8:
880:. Note that the inner sum would be zero for
1606:Tauberian Theory: A Century of Developments
954:provides an explicit representation of the
18:Summation method for some divergent series
1473:
1440:
1429:
1418:
1396:
1387:
1371:
1360:
1347:
1345:
1339:
1328:
1309:
1287:
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1270:
1257:
1247:
1236:
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1203:
1181:
1161:
1148:
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1140:
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1111:
1102:
1096:
1085:
1066:
1050:
1048:
971:
965:
963:
938:
904:
898:
838:
828:
808:
795:
793:
787:
776:
758:
749:
743:
732:
726:
696:
686:
667:
656:
650:
604:
591:
573:
563:
556:
551:
550:
533:
528:
523:
521:
510:
505:
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453:
431:
424:
411:
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403:
392:
376:
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333:
317:
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293:
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285:
274:
255:
233:
227:
216:
203:
193:
182:
177:
169:
164:
159:
71:Learn how and when to remove this message
34:This article includes a list of general
1624:Shawyer, Bruce; Watson, Bruce (1994).
7:
1645:Mathematical Analysis Second Edition
947:{\displaystyle P_{k}(j):=(j+1)^{k}}
1430:
1352:
1282:
1248:
1153:
800:
668:
416:
404:
298:
228:
194:
131:> 0 they are incomparable with
40:it lacks sufficient corresponding
14:
25:
1585:Van Wijngaarden transformation
1570:Abelian and Tauberian theorems
1493:With an appropriate choice of
1306:
1293:
1200:
1187:
1178:
1168:
1014:
1005:
935:
922:
916:
910:
850:
844:
825:
815:
708:
702:
683:
673:
450:
437:
252:
239:
1:
1497:(i.e. equal to or close to −
1628:. Oxford University Press.
1515:) this series converges to
1694:
1647:. Addison Wesley Longman.
1041:Globally convergent series
357:accelerate the convergence
1643:Apostol, Tom M. (1974).
1603:Korevaar, Jacob (2004).
1580:Abel's summation formula
645:= 1 for the formal sum
55:more precise citations.
1483:
1434:
1344:
1286:
1252:
1213:
1145:
1101:
1037:Dirichlet eta function
1021:
948:
893:The particular choice
860:
792:
748:
715:
672:
626:
478:
408:
346:
290:
232:
198:
83:In the mathematics of
1590:Euler–Boole summation
1484:
1414:
1324:
1266:
1232:
1214:
1125:
1081:
1029:Riemann zeta function
1022:
949:
861:
772:
728:
716:
652:
627:
479:
388:
347:
270:
212:
178:
1229:
1047:
962:
897:
725:
649:
494:
369:
158:
1678:Summability methods
1673:Mathematical series
873:is a polynomial of
1575:Abel–Plana formula
1545:Binomial transform
1479:
1209:
1017:
944:
856:
711:
622:
474:
342:
1560:Lambert summation
1467:
1412:
1359:
1322:
1160:
1123:
1079:
994:
956:Bernoulli numbers
807:
770:
611:
466:
423:
305:
268:
81:
80:
73:
1685:
1658:
1639:
1620:
1565:Perron's formula
1555:Cesàro summation
1533:
1531:
1530:
1524:
1521:
1514:
1512:
1511:
1506:
1503:
1488:
1486:
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1023:
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993:
982:
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631:
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623:
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595:
579:
578:
577:
568:
567:
557:
542:
541:
540:
539:
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537:
522:
519:
518:
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516:
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514:
483:
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432:
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415:
407:
402:
387:
386:
351:
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338:
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328:
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312:
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297:
289:
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269:
267:
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234:
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208:
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197:
192:
176:
175:
174:
173:
89:divergent series
76:
69:
65:
62:
56:
51:this article by
42:inline citations
29:
28:
21:
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1663:
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1661:
1655:
1642:
1636:
1623:
1617:
1602:
1598:
1550:Borel summation
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1525:
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436:
410:
372:
367:
366:
329:
313:
292:
251:
238:
199:
165:
161:
156:
155:
143:For some value
141:
125:Borel summation
106:Euler transform
103:
93:Euler summation
77:
66:
60:
57:
47:Please help to
46:
30:
26:
19:
12:
11:
5:
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1454:
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1279:
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1245:
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1239:
1235:
1224:
1206:
1202:
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1128:
1120:
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1094:
1091:
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1084:
1075:
1072:
1069:
1065:
1061:
1058:
1054:
1039:) yields (cf.
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989:
986:
980:
977:
974:
970:
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891:
869:
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841:
837:
831:
827:
823:
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811:
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764:
761:
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746:
741:
738:
735:
731:
710:
707:
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695:
689:
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678:
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662:
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637:
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632:
621:
618:
607:
603:
599:
594:
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586:
583:
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566:
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172:
168:
163:
140:
137:
133:Abel summation
99:
79:
78:
61:September 2012
33:
31:
24:
17:
13:
10:
9:
6:
4:
3:
2:
1690:
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1656:
1654:0-201-00288-4
1650:
1646:
1641:
1637:
1635:0-19-853585-6
1631:
1627:
1622:
1618:
1616:3-540-21058-X
1612:
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1258:
1254:
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1237:
1233:
1225:
1222:
1204:
1196:
1193:
1190:
1182:
1174:
1171:
1157:
1154:
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1136:
1133:
1130:
1126:
1118:
1115:
1112:
1108:
1104:
1097:
1092:
1089:
1086:
1082:
1073:
1070:
1067:
1063:
1059:
1056:
1052:
1042:
1038:
1034:
1030:
1011:
1008:
1002:
999:
996:
990:
987:
984:
978:
975:
972:
968:
957:
939:
931:
928:
925:
919:
913:
905:
901:
892:
888:
884:
879:
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872:
853:
847:
839:
835:
829:
821:
818:
804:
801:
788:
783:
780:
777:
773:
765:
762:
759:
755:
751:
744:
739:
736:
733:
729:
705:
697:
693:
687:
679:
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663:
660:
657:
653:
644:
640:
639:
635:
619:
616:
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592:
588:
584:
581:
574:
570:
564:
560:
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546:
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502:
498:
490:
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471:
468:
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420:
417:
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389:
383:
380:
377:
373:
365:
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363:
360:
358:
339:
334:
330:
324:
321:
318:
314:
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286:
281:
278:
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262:
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223:
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209:
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189:
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162:
154:
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138:
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126:
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118:
113:
111:
107:
102:
98:
94:
90:
86:
75:
72:
64:
54:
50:
44:
43:
37:
32:
23:
22:
16:
1644:
1625:
1609:. Springer.
1605:
1527:
1508:
1494:
1219:which is of
1032:
886:
882:
877:
867:
642:
486:
361:
354:
148:
144:
142:
128:
120:
116:
114:
109:
100:
96:
92:
82:
67:
58:
39:
15:
1221:closed form
53:introducing
1667:Categories
1596:References
139:Definition
85:convergent
36:references
1431:∞
1416:∑
1326:∑
1283:∞
1268:∑
1249:∞
1234:∑
1172:−
1127:∑
1083:∑
1060:−
1009:−
1003:ζ
1000:−
819:−
774:∑
730:∑
677:−
669:∞
654:∑
617:∑
544:∑
405:∞
390:∑
272:∑
229:∞
214:∑
195:∞
180:∑
119:), where
110:Euler sum
104:, if its
1539:See also
958:, since
636:Examples
1532:
1517:
1513:
1499:
721:we get
49:improve
1651:
1632:
1613:
875:degree
641:Using
127:; for
38:, but
1027:(the
885:>
1649:ISBN
1630:ISBN
1611:ISBN
1526:1 −
87:and
866:if
1669::
1043:)
920::=
472:1.
210::=
135:.
91:,
1657:.
1638:.
1619:.
1534:.
1528:z
1523:/
1520:1
1509:z
1505:/
1502:1
1495:y
1475:i
1470:)
1464:y
1461:+
1458:1
1453:z
1450:y
1447:+
1444:1
1438:(
1426:0
1423:=
1420:i
1409:y
1406:+
1403:1
1399:y
1394:=
1389:j
1385:z
1379:1
1376:+
1373:j
1369:y
1362:)
1357:j
1354:i
1349:(
1341:i
1336:0
1333:=
1330:j
1317:1
1314:+
1311:i
1307:)
1303:y
1300:+
1297:1
1294:(
1290:1
1278:0
1275:=
1272:i
1264:=
1259:j
1255:z
1244:0
1241:=
1238:j
1223:.
1205:k
1201:)
1197:1
1194:+
1191:j
1188:(
1183:j
1179:)
1175:1
1169:(
1163:)
1158:j
1155:i
1150:(
1142:i
1137:0
1134:=
1131:j
1119:1
1116:+
1113:i
1109:2
1105:1
1098:k
1093:0
1090:=
1087:i
1074:1
1071:+
1068:k
1064:2
1057:1
1053:1
1033:k
1015:)
1012:k
1006:(
997:=
991:1
988:+
985:k
979:1
976:+
973:k
969:B
940:k
936:)
932:1
929:+
926:j
923:(
917:)
914:j
911:(
906:k
902:P
887:k
883:i
878:k
870:k
868:P
854:,
851:)
848:j
845:(
840:k
836:P
830:j
826:)
822:1
816:(
810:)
805:j
802:i
797:(
789:i
784:0
781:=
778:j
766:1
763:+
760:i
756:2
752:1
745:k
740:0
737:=
734:i
709:)
706:j
703:(
698:k
694:P
688:j
684:)
680:1
674:(
664:0
661:=
658:j
643:y
620:.
606:2
602:y
598:+
593:1
589:y
585:+
582:1
575:2
571:y
565:1
561:y
554:E
547:=
535:2
531:y
526:E
512:1
508:y
503:E
469:=
461:1
458:+
455:i
451:)
447:y
444:+
441:1
438:(
434:1
426:)
421:j
418:i
413:(
400:j
397:=
394:i
384:1
381:+
378:j
374:y
340:.
335:j
331:a
325:1
322:+
319:j
315:y
308:)
303:j
300:i
295:(
287:i
282:0
279:=
276:j
263:1
260:+
257:i
253:)
249:y
246:+
243:1
240:(
236:1
224:0
221:=
218:i
205:j
201:a
190:0
187:=
184:j
171:y
167:E
149:y
145:y
129:q
121:q
117:q
101:n
97:a
74:)
68:(
63:)
59:(
45:.
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