422:
322:
440:
943:
Probably the most interesting part of this theorem is that the Cauchy condition implies the existence of the limit: this is indeed related to the completeness of the real line. The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing
701:
We can use the results about convergence of the sequence of partial sums of the infinite series and apply them to the convergence of the infinite series itself. The Cauchy
Criterion test is one such application. For any real sequence
938:
251:
689:), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to
629:
781:
102:
810:
132:
687:
665:
546:
520:
364:
306:
280:
1083:
727:
391:
411:
155:
461:
834:
1017:
427:(b) A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses.
163:
1042:
487:
465:
413:. If the space containing the sequence is complete, the "ultimate destination" of this sequence (that is, the limit) exists.
1069:
1064:
450:
1103:
1077:
469:
454:
1059:
573:
738:
59:
20:
642:
561:
39:
953:
789:
111:
553:
31:
670:
648:
529:
503:
690:
421:
44:
321:
38:. It relies on bounding sums of terms in the series. This convergence criterion is named after
1038:
1013:
333:
35:
285:
259:
705:
369:
1009:
730:
635:
557:
549:
396:
140:
135:
105:
1097:
983:
933:{\displaystyle |s_{m}-s_{n}|=\left|\sum _{k=n+1}^{m}a_{k}\right|<\varepsilon .}
565:
523:
439:
949:
948:
This article incorporates material from Cauchy criterion for convergence on
328:
975:
246:{\displaystyle |a_{n+1}+a_{n+2}+\cdots +a_{n+p}|<\varepsilon }
1058:
Kudryavtsev, Lev D.; De Lellis, Camillo; Artemisfowl3rd (2013).
433:
564:. From here, the series is convergent if and only if the
944:
oscillation condition is equivalent to convergence".
837:
792:
741:
708:
673:
651:
576:
532:
506:
399:
372:
336:
288:
262:
166:
143:
114:
62:
1008:. Undergraduate Texts in Mathematics. New York, NY:
932:
804:
775:
729:, the above results on convergence imply that the
721:
681:
659:
623:
540:
514:
405:
385:
358:
300:
274:
245:
149:
126:
96:
954:Creative Commons Attribution/Share-Alike License
976:"Answer to 'Origin of Cauchy convergence test'"
641:Cauchy's convergence test can only be used in
8:
1082:: CS1 maint: numeric names: authors list (
624:{\displaystyle s_{n}:=\sum _{i=0}^{n}a_{i}}
468:. Unsourced material may be challenged and
776:{\displaystyle \sum _{k=1}^{\infty }a_{k}}
97:{\displaystyle \sum _{i=0}^{\infty }a_{i}}
1037:. Upper Saddle River, NJ: Prentice Hall.
910:
900:
883:
866:
860:
847:
838:
836:
791:
767:
757:
746:
740:
713:
707:
675:
674:
672:
653:
652:
650:
615:
605:
594:
581:
575:
534:
533:
531:
508:
507:
505:
488:Learn how and when to remove this message
398:
377:
371:
344:
335:
287:
261:
232:
220:
195:
176:
167:
165:
142:
113:
88:
78:
67:
61:
966:
1075:
7:
466:adding citations to reliable sources
786:converges if and only if for every
980:History of Science and Mathematics
758:
79:
14:
805:{\displaystyle \varepsilon >0}
500:The test works because the space
127:{\displaystyle \varepsilon >0}
42:who published it in his textbook
438:
420:
320:
952:, which is licensed under the
867:
839:
350:
337:
233:
168:
1:
1070:European Mathematical Society
16:Criterion for infinite series
682:{\displaystyle \mathbb {C} }
660:{\displaystyle \mathbb {R} }
541:{\displaystyle \mathbb {C} }
515:{\displaystyle \mathbb {R} }
1065:Encyclopedia of Mathematics
1035:An Introduction to Analysis
1120:
18:
1062:. In Rehmann, Ulf (ed.).
327:(a) The plot of a Cauchy
30:is a method used to test
1004:Abbott, Stephen (2001).
359:{\displaystyle (x_{n}),}
21:Cauchy condensation test
19:Not to be confused with
301:{\displaystyle p\geq 1}
28:Cauchy convergence test
1033:Wade, William (2010).
1006:Understanding analysis
934:
905:
806:
777:
762:
723:
693:the series converges.
683:
661:
643:complete metric spaces
625:
610:
542:
516:
407:
387:
360:
302:
276:
275:{\displaystyle n>N}
247:
151:
128:
98:
83:
935:
879:
807:
778:
742:
724:
722:{\displaystyle a_{k}}
684:
662:
626:
590:
543:
517:
408:
388:
386:{\displaystyle x_{n}}
361:
303:
277:
248:
152:
129:
99:
63:
40:Augustin-Louis Cauchy
835:
790:
739:
706:
671:
649:
574:
530:
504:
462:improve this section
397:
370:
334:
286:
260:
164:
141:
112:
60:
974:Allegranza, Mauro.
930:
812:there is a number
802:
773:
719:
679:
657:
621:
538:
512:
403:
383:
366:shown in blue, as
356:
298:
272:
243:
147:
124:
94:
1104:Convergence tests
1078:cite encyclopedia
1060:"Cauchy criteria"
1019:978-0-387-21506-8
498:
497:
490:
406:{\displaystyle n}
150:{\displaystyle N}
1111:
1088:
1087:
1081:
1073:
1055:
1049:
1048:
1030:
1024:
1023:
1001:
995:
994:
992:
990:
971:
939:
937:
936:
931:
920:
916:
915:
914:
904:
899:
870:
865:
864:
852:
851:
842:
811:
809:
808:
803:
782:
780:
779:
774:
772:
771:
761:
756:
728:
726:
725:
720:
718:
717:
688:
686:
685:
680:
678:
666:
664:
663:
658:
656:
630:
628:
627:
622:
620:
619:
609:
604:
586:
585:
547:
545:
544:
539:
537:
521:
519:
518:
513:
511:
493:
486:
482:
479:
473:
442:
434:
424:
412:
410:
409:
404:
392:
390:
389:
384:
382:
381:
365:
363:
362:
357:
349:
348:
324:
307:
305:
304:
299:
281:
279:
278:
273:
252:
250:
249:
244:
236:
231:
230:
206:
205:
187:
186:
171:
156:
154:
153:
148:
133:
131:
130:
125:
103:
101:
100:
95:
93:
92:
82:
77:
1119:
1118:
1114:
1113:
1112:
1110:
1109:
1108:
1094:
1093:
1092:
1091:
1074:
1057:
1056:
1052:
1045:
1032:
1031:
1027:
1020:
1010:Springer Verlag
1003:
1002:
998:
988:
986:
973:
972:
968:
963:
906:
878:
874:
856:
843:
833:
832:
788:
787:
763:
737:
736:
731:infinite series
709:
704:
703:
699:
669:
668:
647:
646:
636:Cauchy sequence
611:
577:
572:
571:
550:complex numbers
528:
527:
502:
501:
494:
483:
477:
474:
459:
443:
432:
431:
430:
429:
428:
425:
416:
415:
414:
395:
394:
373:
368:
367:
340:
332:
331:
325:
314:
284:
283:
258:
257:
216:
191:
172:
162:
161:
139:
138:
110:
109:
84:
58:
57:
54:
45:Cours d'Analyse
32:infinite series
24:
17:
12:
11:
5:
1117:
1115:
1107:
1106:
1096:
1095:
1090:
1089:
1050:
1043:
1025:
1018:
1012:. p. 63.
996:
965:
964:
962:
959:
941:
940:
929:
926:
923:
919:
913:
909:
903:
898:
895:
892:
889:
886:
882:
877:
873:
869:
863:
859:
855:
850:
846:
841:
801:
798:
795:
784:
783:
770:
766:
760:
755:
752:
749:
745:
716:
712:
698:
695:
677:
655:
632:
631:
618:
614:
608:
603:
600:
597:
593:
589:
584:
580:
558:absolute value
536:
526:and the space
510:
496:
495:
446:
444:
437:
426:
419:
418:
417:
402:
380:
376:
355:
352:
347:
343:
339:
326:
319:
318:
317:
316:
315:
313:
310:
297:
294:
291:
271:
268:
265:
256:holds for all
254:
253:
242:
239:
235:
229:
226:
223:
219:
215:
212:
209:
204:
201:
198:
194:
190:
185:
182:
179:
175:
170:
146:
136:natural number
123:
120:
117:
106:if and only if
104:is convergent
91:
87:
81:
76:
73:
70:
66:
53:
50:
15:
13:
10:
9:
6:
4:
3:
2:
1116:
1105:
1102:
1101:
1099:
1085:
1079:
1071:
1067:
1066:
1061:
1054:
1051:
1046:
1044:9780132296380
1040:
1036:
1029:
1026:
1021:
1015:
1011:
1007:
1000:
997:
985:
984:StackExchange
981:
977:
970:
967:
960:
958:
957:
955:
951:
945:
927:
924:
921:
917:
911:
907:
901:
896:
893:
890:
887:
884:
880:
875:
871:
861:
857:
853:
848:
844:
831:
830:
829:
827:
823:
819:
815:
799:
796:
793:
768:
764:
753:
750:
747:
743:
735:
734:
733:
732:
714:
710:
696:
694:
692:
644:
639:
637:
616:
612:
606:
601:
598:
595:
591:
587:
582:
578:
570:
569:
568:
567:
563:
559:
556:given by the
555:
551:
525:
492:
489:
481:
478:February 2022
471:
467:
463:
457:
456:
452:
447:This section
445:
441:
436:
435:
423:
400:
378:
374:
353:
345:
341:
330:
323:
311:
309:
295:
292:
289:
269:
266:
263:
240:
237:
227:
224:
221:
217:
213:
210:
207:
202:
199:
196:
192:
188:
183:
180:
177:
173:
160:
159:
158:
144:
137:
121:
118:
115:
107:
89:
85:
74:
71:
68:
64:
51:
49:
47:
46:
41:
37:
33:
29:
22:
1068:. Springer,
1063:
1053:
1034:
1028:
1005:
999:
989:10 September
987:. Retrieved
979:
969:
947:
946:
942:
825:
821:
817:
816:, such that
813:
785:
700:
640:
633:
566:partial sums
524:real numbers
499:
484:
475:
460:Please help
448:
255:
55:
43:
27:
25:
560:) are both
312:Explanation
134:there is a
36:convergence
961:References
950:PlanetMath
552:(with the
157:such that
108:for every
925:ε
881:∑
854:−
794:ε
759:∞
744:∑
645:(such as
592:∑
449:does not
293:≥
241:ε
211:⋯
116:ε
80:∞
65:∑
56:A series
52:Statement
1098:Category
562:complete
329:sequence
282:and all
470:removed
455:sources
393:versus
1041:
1016:
828:imply
634:are a
554:metric
48:1821.
697:Proof
691:prove
1084:link
1039:ISBN
1014:ISBN
991:2021
922:<
797:>
667:and
453:any
451:cite
267:>
238:<
119:>
34:for
26:The
548:of
522:of
464:by
1100::
1080:}}
1076:{{
982:.
978:.
824:≥
820:≥
638:.
588::=
308:.
1086:)
1072:.
1047:.
1022:.
993:.
956:.
928:.
918:|
912:k
908:a
902:m
897:1
894:+
891:n
888:=
885:k
876:|
872:=
868:|
862:n
858:s
849:m
845:s
840:|
826:N
822:n
818:m
814:N
800:0
769:k
765:a
754:1
751:=
748:k
715:k
711:a
676:C
654:R
617:i
613:a
607:n
602:0
599:=
596:i
583:n
579:s
535:C
509:R
491:)
485:(
480:)
476:(
472:.
458:.
401:n
379:n
375:x
354:,
351:)
346:n
342:x
338:(
296:1
290:p
270:N
264:n
234:|
228:p
225:+
222:n
218:a
214:+
208:+
203:2
200:+
197:n
193:a
189:+
184:1
181:+
178:n
174:a
169:|
145:N
122:0
90:i
86:a
75:0
72:=
69:i
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
