Theorem cvgratnn 12221

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 12222 and cvgratgt0 12223, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 12039 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 9893	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9606	. . 3 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10849	. 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 10029	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rprecred 10044	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
9		1red 8291	. . . . . . . 8 |- ( ph -> 1 e. RR )
10	8, 9	resubcld 8656	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
11		cvgratnn.4	. . . . . . . . 9 |- ( ph -> A < 1 )
12	7	reclt1d 10046	. . . . . . . . 9 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
13	11, 12	mpbid 147	. . . . . . . 8 |- ( ph -> 1 < ( 1 / A ) )
14	9, 8	posdifd 8808	. . . . . . . 8 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
15	13, 14	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
16	10, 15	elrpd 10029	. . . . . 6 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
17	16	rpreccld 10043	. . . . 5 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
18	17, 7	rpdivcld 10050	. . . 4 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
19		fveq2 5672	. . . . . . . 8 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
20	19	eleq1d 2303	. . . . . . 7 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
21	3	ralrimiva 2617	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
22		1nn 9250	. . . . . . . 8 |- 1 e. NN
23	22	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
24	20, 21, 23	rspcdva 2928	. . . . . 6 |- ( ph -> ( F ` 1 ) e. CC )
25	24	abscld 11870	. . . . 5 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
26	24	absge0d 11873	. . . . 5 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
27	25, 26	ge0p1rpd 10063	. . . 4 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
28	18, 27	rpmulcld 10049	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
29	9, 5	resubcld 8656	. . . . 5 |- ( ph -> ( 1 - A ) e. RR )
30	5, 9	posdifd 8808	. . . . . 6 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
31	11, 30	mpbid 147	. . . . 5 |- ( ph -> 0 < ( 1 - A ) )
32	29, 31	elrpd 10029	. . . 4 |- ( ph -> ( 1 - A ) e. RR+ )
33	7, 32	rpdivcld 10050	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
34	28, 33	rpmulcld 10049	. 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 531	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> m e. NN )
42		simprr 533	. . . 4 |- ( ( ph /\ ( m e. NN /\ n e. ( ZZ>= ` m ) ) ) -> n e. ( ZZ>= ` m ) )
43	35, 36, 37, 38, 40, 41, 42	cvgratnnlemrate 12220	. . 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 2626	. 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 12039	1 |- ( ph -> seq 1 ( + , F ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4111   dom cdm 4751   ` cfv 5354  (class class class)co 6052   CCcc 8127   RRcr 8128   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   < clt 8310   <_ cle 8311   - cmin 8446   / cdiv 8948   NNcn 9239   ZZ>=cuz 9856   seqcseq 10813   abscabs 11686   ~~> cli 11967

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-ico 10230  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043

This theorem is referenced by:  cvgratz  12222