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Mirrors > Home > ILE Home > Th. List > cvgratgt0 | Unicode version |
Description: Ratio test for convergence of a complex infinite series. If the ratio of the absolute values of successive terms in an infinite sequence is less than 1 for all terms beyond some index , then the infinite sum of the terms of converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.) |
Ref | Expression |
---|---|
cvgrat.1 | |
cvgrat.2 | |
cvgrat.3 | |
cvgrat.4 | |
cvgrat.gt0 | |
cvgrat.5 | |
cvgrat.6 | |
cvgrat.7 |
Ref | Expression |
---|---|
cvgratgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvgrat.2 | . . 3 | |
2 | cvgrat.5 | . . . 4 | |
3 | eluzelz 9340 | . . . . 5 | |
4 | cvgrat.1 | . . . . 5 | |
5 | 3, 4 | eleq2s 2234 | . . . 4 |
6 | 2, 5 | syl 14 | . . 3 |
7 | cvgrat.3 | . . 3 | |
8 | cvgrat.4 | . . 3 | |
9 | cvgrat.gt0 | . . 3 | |
10 | 1 | eleq2i 2206 | . . . . . . 7 |
11 | 10 | biimpi 119 | . . . . . 6 |
12 | 2, 4 | eleqtrdi 2232 | . . . . . 6 |
13 | uztrn 9347 | . . . . . 6 | |
14 | 11, 12, 13 | syl2anr 288 | . . . . 5 |
15 | 14, 4 | eleqtrrdi 2233 | . . . 4 |
16 | cvgrat.6 | . . . 4 | |
17 | 15, 16 | syldan 280 | . . 3 |
18 | cvgrat.7 | . . 3 | |
19 | 1, 6, 7, 8, 9, 17, 18 | cvgratz 11306 | . 2 |
20 | 4, 2, 16 | iserex 11113 | . 2 |
21 | 19, 20 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cdm 4539 cfv 5123 (class class class)co 5774 cc 7623 cr 7624 cc0 7625 c1 7626 caddc 7628 cmul 7630 clt 7805 cle 7806 cz 9059 cuz 9331 cseq 10223 cabs 10774 cli 11052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-mulrcl 7724 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-precex 7735 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 ax-pre-mulgt0 7742 ax-pre-mulext 7743 ax-arch 7744 ax-caucvg 7745 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-reap 8342 df-ap 8349 df-div 8438 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-n0 8983 df-z 9060 df-uz 9332 df-q 9417 df-rp 9447 df-ico 9682 df-fz 9796 df-fzo 9925 df-seqfrec 10224 df-exp 10298 df-ihash 10527 df-cj 10619 df-re 10620 df-im 10621 df-rsqrt 10775 df-abs 10776 df-clim 11053 df-sumdc 11128 |
This theorem is referenced by: efcllemp 11369 |
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