Theorem cvgratnn 11842

Description: Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms, then the infinite sum of the terms of F converges to a complex number. Although this theorem is similar to cvgratz 11843 and cvgratgt0 11844, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11661 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)

Hypotheses

Ref	Expression
cvgratnn.3	|- ( ph -> A e. RR )
cvgratnn.4	|- ( ph -> A < 1 )
cvgratnn.gt0	|- ( ph -> 0 < A )
cvgratnn.6	|- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
cvgratnn.7	|- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )

Assertion

Ref	Expression
cvgratnn	|- ( ph -> seq 1 ( + , F ) e. dom ~~> )

Distinct variable groups:   A, k   k, F   ph, k

Proof of Theorem cvgratnn

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 9684	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9399	. . 3 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10628	. 2 |- ( ph -> seq 1 ( + , F ) : NN --> CC )
5		cvgratnn.3	. . . . . . . . . 10 |- ( ph -> A e. RR )
6		cvgratnn.gt0	. . . . . . . . . 10 |- ( ph -> 0 < A )
7	5, 6	elrpd 9815	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rprecred 9830	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
9		1red 8087	. . . . . . . 8 |- ( ph -> 1 e. RR )
10	8, 9	resubcld 8453	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
11		cvgratnn.4	. . . . . . . . 9 |- ( ph -> A < 1 )
12	7	reclt1d 9832	. . . . . . . . 9 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
13	11, 12	mpbid 147	. . . . . . . 8 |- ( ph -> 1 < ( 1 / A ) )
14	9, 8	posdifd 8605	. . . . . . . 8 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
15	13, 14	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
16	10, 15	elrpd 9815	. . . . . 6 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
17	16	rpreccld 9829	. . . . 5 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
18	17, 7	rpdivcld 9836	. . . 4 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
19		fveq2 5576	. . . . . . . 8 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
20	19	eleq1d 2274	. . . . . . 7 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
21	3	ralrimiva 2579	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
22		1nn 9047	. . . . . . . 8 |- 1 e. NN
23	22	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
24	20, 21, 23	rspcdva 2882	. . . . . 6 |- ( ph -> ( F ` 1 ) e. CC )
25	24	abscld 11492	. . . . 5 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
26	24	absge0d 11495	. . . . 5 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
27	25, 26	ge0p1rpd 9849	. . . 4 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
28	18, 27	rpmulcld 9835	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
29	9, 5	resubcld 8453	. . . . 5 |- ( ph -> ( 1 - A ) e. RR )
30	5, 9	posdifd 8605	. . . . . 6 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
31	11, 30	mpbid 147	. . . . 5 |- ( ph -> 0 < ( 1 - A ) )
32	29, 31	elrpd 9815	. . . 4 |- ( ph -> ( 1 - A ) e. RR+ )
33	7, 32	rpdivcld 9836	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
34	28, 33	rpmulcld 9835	. 2 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) e. RR+ )
35	5	adantr 276	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> A e. RR )
36	11	adantr 276	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> A < 1 )
37	6	adantr 276	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> 0 < A )
38	3	adantlr 477	. . . 4 |- ( ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) /\ k e. NN ) -> ( F ` k ) e. CC )
39		cvgratnn.7	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
40	39	adantlr 477	. . . 4 |- ( ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
41		simprl 529	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> m e. NN )
42		simprr 531	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> n e. ( ZZ>= ` m ) )
43	35, 36, 37, 38, 40, 41, 42	cvgratnnlemrate 11841	. . 3 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> ( abs ` ( ( seq 1 ( + , F ) ` n ) - ( seq 1 ( + , F ) ` m ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / m ) )
44	43	ralrimivva 2588	. 2 |- ( ph -> A. m e. NN A. n e. ( ZZ>= ` m ) ( abs ` ( ( seq 1 ( + , F ) ` n ) - ( seq 1 ( + , F ) ` m ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / m ) )
45	4, 34, 44	climcvg1n 11661	1 |- ( ph -> seq 1 ( + , F ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4044   dom cdm 4675   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   - cmin 8243   / cdiv 8745   NNcn 9036   ZZ>=cuz 9648   seqcseq 10592   abscabs 11308   ~~> cli 11589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-ico 10016  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  cvgratz  11843