Theorem cvgratnn 12298

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 12299 and cvgratgt0 12300, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 12116 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 9958	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9671	. . 3 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10920	. 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 10094	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rprecred 10109	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
9		1red 8341	. . . . . . . 8 |- ( ph -> 1 e. RR )
10	8, 9	resubcld 8708	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
11		cvgratnn.4	. . . . . . . . 9 |- ( ph -> A < 1 )
12	7	reclt1d 10111	. . . . . . . . 9 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
13	11, 12	mpbid 147	. . . . . . . 8 |- ( ph -> 1 < ( 1 / A ) )
14	9, 8	posdifd 8860	. . . . . . . 8 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
15	13, 14	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
16	10, 15	elrpd 10094	. . . . . 6 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
17	16	rpreccld 10108	. . . . 5 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
18	17, 7	rpdivcld 10115	. . . 4 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
19		fveq2 5695	. . . . . . . 8 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
20	19	eleq1d 2307	. . . . . . 7 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
21	3	ralrimiva 2623	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
22		1nn 9315	. . . . . . . 8 |- 1 e. NN
23	22	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
24	20, 21, 23	rspcdva 2934	. . . . . 6 |- ( ph -> ( F ` 1 ) e. CC )
25	24	abscld 11947	. . . . 5 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
26	24	absge0d 11950	. . . . 5 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
27	25, 26	ge0p1rpd 10128	. . . 4 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
28	18, 27	rpmulcld 10114	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
29	9, 5	resubcld 8708	. . . . 5 |- ( ph -> ( 1 - A ) e. RR )
30	5, 9	posdifd 8860	. . . . . 6 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
31	11, 30	mpbid 147	. . . . 5 |- ( ph -> 0 < ( 1 - A ) )
32	29, 31	elrpd 10094	. . . 4 |- ( ph -> ( 1 - A ) e. RR+ )
33	7, 32	rpdivcld 10115	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
34	28, 33	rpmulcld 10114	. 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 481	. . . 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 481	. . . 4 |- ( ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
41		simprl 535	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> m e. NN )
42		simprr 537	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> n e. ( ZZ>= ` m ) )
43	35, 36, 37, 38, 40, 41, 42	cvgratnnlemrate 12297	. . 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 2632	. 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 12116	1 |- ( ph -> seq 1 ( + , F ) e. dom ~~> )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9002   NNcn 9304   ZZ>=cuz 9921   seqcseq 10884   abscabs 11763   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ico 10296  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  cvgratz  12299