686:
723:
of the infinite expression may be ambiguous or not well-defined; for instance, there are multiple summation rules available for assigning values to series, and the same series may have different values according to different summation rules if the series is not
445:
272:
425:
164:
343:
47:, or in which the nesting of the operators continues to an infinite depth. A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions (much like a
681:{\displaystyle c_{0}+{\underset {n=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{c_{n}}}=c_{0}+{\cfrac {1}{c_{1}+{\cfrac {1}{c_{2}+{\cfrac {1}{c_{3}+{\cfrac {1}{c_{4}+\ddots }}}}}}}},}
184:
757:
843:
363:
896:
904:
851:
75:
292:
931:
697:
926:
737:
36:
40:
44:
725:
693:
888:
713:
709:
868:
433:
17:
900:
847:
816:
808:
747:
798:
705:
172:
835:
762:
280:
267:{\displaystyle \prod _{n=0}^{\infty }b_{n}=b_{0}\times b_{1}\times b_{2}\times \cdots }
920:
742:
48:
803:
786:
752:
63:
420:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdot ^{\cdot ^{\cdot }}}}}}
51:), but there are several instances of infinite expressions that are well-defined.
782:
28:
812:
351:
867:
Moroni, Luca (2019). "The strange properties of the infinite power tower".
820:
159:{\displaystyle \sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots \,}
873:
842:(Hardcover). J.D. Blanton (translator). Springer Verlag. p.
338:{\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}
634:
622:
601:
589:
568:
556:
535:
523:
637:
625:
604:
592:
571:
559:
538:
526:
448:
366:
295:
187:
78:
895:(Hardcover). American Mathematical Society. p.
787:"The syntax of a language with infinite expressions"
680:
419:
337:
266:
158:
59:Examples of well-defined infinite expressions are
840:Introduction to Analysis of the Infinite, Book I
719:Even for well-defined infinite expressions, the
791:Bulletin of the American Mathematical Society
8:
758:Infinite compositions of analytic functions
872:
802:
643:
638:
626:
619:
610:
605:
593:
586:
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511:
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487:
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368:
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318:
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296:
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155:
143:
130:
117:
104:
94:
83:
77:
774:
893:Analytic Theory of Continued Fractions
7:
470:
466:
204:
95:
25:
18:Infinite expression (mathematics)
804:10.1090/S0002-9904-1938-06672-4
692:where the left hand side uses
1:
434:infinite continued fractions
43:take an infinite number of
948:
738:Iterated binary operation
281:infinite nested radicals
708:, one can use infinite
682:
421:
339:
268:
208:
160:
99:
932:Mathematical analysis
726:absolutely convergent
683:
422:
352:infinite power towers
340:
269:
188:
161:
79:
889:Wall, Hubert Stanley
838:(November 1, 1988).
698:Kettenbruch notation
446:
364:
293:
185:
76:
636:
624:
603:
591:
570:
558:
537:
525:
33:infinite expression
891:(March 28, 2000).
678:
671:
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661:
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631:
598:
565:
532:
485:
417:
335:
264:
156:
906:978-0-8218-2106-0
853:978-0-387-96824-7
748:Decimal expansion
673:
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557:
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524:
502:
473:
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333:
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173:infinite products
16:(Redirected from
939:
927:Abstract algebra
911:
910:
885:
879:
878:
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864:
858:
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832:
826:
824:
806:
785:(January 1938).
779:
706:infinitary logic
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122:
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109:
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98:
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21:
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865:
861:
854:
836:Euler, Leonhard
834:
833:
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771:
734:
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621:
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573:
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367:
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291:
290:
248:
235:
222:
209:
183:
182:
139:
126:
113:
100:
74:
73:
57:
49:set of all sets
23:
22:
15:
12:
11:
5:
945:
943:
935:
934:
929:
919:
918:
913:
912:
905:
880:
859:
852:
827:
773:
772:
770:
767:
766:
765:
763:Omega language
760:
755:
750:
745:
740:
733:
730:
702:
701:
690:
689:
688:
677:
654:
651:
646:
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629:
618:
613:
609:
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585:
580:
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563:
552:
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543:
530:
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510:
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483:
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477:
472:
468:
461:
456:
452:
438:
437:
430:
429:
428:
427:
406:
402:
397:
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381:
372:
356:
355:
348:
347:
346:
345:
328:
325:
322:
317:
314:
311:
306:
303:
300:
285:
284:
277:
276:
275:
274:
263:
260:
255:
251:
247:
242:
238:
234:
229:
225:
221:
216:
212:
206:
201:
198:
195:
191:
177:
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169:
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167:
166:
154:
151:
146:
142:
138:
133:
129:
125:
120:
116:
112:
107:
103:
97:
92:
89:
86:
82:
68:
67:
56:
53:
39:in which some
24:
14:
13:
10:
9:
6:
4:
3:
2:
944:
933:
930:
928:
925:
924:
922:
908:
902:
898:
894:
890:
884:
881:
875:
870:
863:
860:
855:
849:
845:
841:
837:
831:
828:
822:
818:
814:
810:
805:
800:
796:
792:
788:
784:
778:
775:
768:
764:
761:
759:
756:
754:
751:
749:
746:
744:
743:Infinite word
741:
739:
736:
735:
731:
729:
727:
722:
717:
715:
712:and infinite
711:
707:
699:
695:
691:
675:
652:
649:
644:
640:
627:
616:
611:
607:
594:
583:
578:
574:
561:
550:
545:
541:
528:
517:
512:
508:
504:
497:
493:
489:
481:
478:
475:
459:
454:
450:
442:
441:
440:
439:
435:
432:
431:
404:
400:
395:
388:
379:
370:
360:
359:
358:
357:
353:
350:
349:
326:
323:
320:
315:
312:
309:
304:
301:
298:
289:
288:
287:
286:
282:
279:
278:
261:
258:
253:
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236:
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227:
223:
219:
214:
210:
199:
196:
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171:
170:
152:
149:
144:
140:
136:
131:
127:
123:
118:
114:
110:
105:
101:
90:
87:
84:
80:
72:
71:
70:
69:
65:
64:infinite sums
62:
61:
60:
54:
52:
50:
46:
42:
38:
34:
30:
19:
892:
883:
862:
839:
830:
797:(1): 33–34.
794:
793:(Abstract).
790:
783:Helmer, Olaf
777:
753:Power series
720:
718:
714:disjunctions
710:conjunctions
703:
58:
32:
26:
29:mathematics
921:Categories
874:1908.05559
769:References
37:expression
813:0002-9904
653:⋱
471:∞
436:, such as
405:⋅
401:⋅
396:⋅
354:, such as
327:⋯
283:, such as
262:⋯
259:×
246:×
233:×
205:∞
190:∏
175:, such as
153:⋯
96:∞
81:∑
66:, such as
45:arguments
41:operators
732:See also
55:Examples
821:5797393
903:
850:
819:
811:
35:is an
869:arXiv
721:value
694:Gauss
31:, an
901:ISBN
848:ISBN
817:OCLC
809:ISSN
844:303
799:doi
704:In
696:'s
27:In
923::
899:.
897:14
846:.
815:.
807:.
795:44
789:.
728:.
716:.
909:.
877:.
871::
856:.
825:.
823:.
801::
700:.
676:,
650:+
645:4
641:c
628:1
617:+
612:3
608:c
595:1
584:+
579:2
575:c
562:1
551:+
546:1
542:c
529:1
518:+
513:0
509:c
505:=
498:n
494:c
490:1
482:1
479:=
476:n
467:K
460:+
455:0
451:c
389:2
380:2
371:2
324:+
321:1
316:3
313:+
310:1
305:2
302:+
299:1
254:2
250:b
241:1
237:b
228:0
224:b
220:=
215:n
211:b
200:0
197:=
194:n
150:+
145:2
141:a
137:+
132:1
128:a
124:+
119:0
115:a
111:=
106:n
102:a
91:0
88:=
85:n
20:)
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