Theorem cvgratnn 11472

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 11473 and cvgratgt0 11474, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11291 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 9501	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9218	. . 3 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10409	. 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 9629	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rprecred 9644	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
9		1red 7914	. . . . . . . 8 |- ( ph -> 1 e. RR )
10	8, 9	resubcld 8279	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
11		cvgratnn.4	. . . . . . . . 9 |- ( ph -> A < 1 )
12	7	reclt1d 9646	. . . . . . . . 9 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
13	11, 12	mpbid 146	. . . . . . . 8 |- ( ph -> 1 < ( 1 / A ) )
14	9, 8	posdifd 8430	. . . . . . . 8 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
15	13, 14	mpbid 146	. . . . . . 7 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
16	10, 15	elrpd 9629	. . . . . 6 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
17	16	rpreccld 9643	. . . . 5 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
18	17, 7	rpdivcld 9650	. . . 4 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
19		fveq2 5486	. . . . . . . 8 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
20	19	eleq1d 2235	. . . . . . 7 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
21	3	ralrimiva 2539	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
22		1nn 8868	. . . . . . . 8 |- 1 e. NN
23	22	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
24	20, 21, 23	rspcdva 2835	. . . . . 6 |- ( ph -> ( F ` 1 ) e. CC )
25	24	abscld 11123	. . . . 5 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
26	24	absge0d 11126	. . . . 5 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
27	25, 26	ge0p1rpd 9663	. . . 4 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
28	18, 27	rpmulcld 9649	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
29	9, 5	resubcld 8279	. . . . 5 |- ( ph -> ( 1 - A ) e. RR )
30	5, 9	posdifd 8430	. . . . . 6 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
31	11, 30	mpbid 146	. . . . 5 |- ( ph -> 0 < ( 1 - A ) )
32	29, 31	elrpd 9629	. . . 4 |- ( ph -> ( 1 - A ) e. RR+ )
33	7, 32	rpdivcld 9650	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
34	28, 33	rpmulcld 9649	. 2 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) e. RR+ )
35	5	adantr 274	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> A e. RR )
36	11	adantr 274	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> A < 1 )
37	6	adantr 274	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> 0 < A )
38	3	adantlr 469	. . . 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 469	. . . 4 |- ( ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
41		simprl 521	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> m e. NN )
42		simprr 522	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> n e. ( ZZ>= ` m ) )
43	35, 36, 37, 38, 40, 41, 42	cvgratnnlemrate 11471	. . 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 2548	. 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 11291	1 |- ( ph -> seq 1 ( + , F ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   class class class wbr 3982   dom cdm 4604   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   <_ cle 7934   - cmin 8069   / cdiv 8568   NNcn 8857   ZZ>=cuz 9466   seqcseq 10380   abscabs 10939   ~~> cli 11219

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by:  cvgratz  11473