Theorem cvgratnn 12042

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 12043 and cvgratgt0 12044, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11861 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 9758	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9473	. . 3 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10705	. 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 9889	. . . . . . . . 9 |- ( ph -> A e. RR+ )
8	7	rprecred 9904	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
9		1red 8161	. . . . . . . 8 |- ( ph -> 1 e. RR )
10	8, 9	resubcld 8527	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
11		cvgratnn.4	. . . . . . . . 9 |- ( ph -> A < 1 )
12	7	reclt1d 9906	. . . . . . . . 9 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
13	11, 12	mpbid 147	. . . . . . . 8 |- ( ph -> 1 < ( 1 / A ) )
14	9, 8	posdifd 8679	. . . . . . . 8 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
15	13, 14	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
16	10, 15	elrpd 9889	. . . . . 6 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
17	16	rpreccld 9903	. . . . 5 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
18	17, 7	rpdivcld 9910	. . . 4 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
19		fveq2 5627	. . . . . . . 8 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
20	19	eleq1d 2298	. . . . . . 7 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
21	3	ralrimiva 2603	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
22		1nn 9121	. . . . . . . 8 |- 1 e. NN
23	22	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
24	20, 21, 23	rspcdva 2912	. . . . . 6 |- ( ph -> ( F ` 1 ) e. CC )
25	24	abscld 11692	. . . . 5 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
26	24	absge0d 11695	. . . . 5 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
27	25, 26	ge0p1rpd 9923	. . . 4 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
28	18, 27	rpmulcld 9909	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
29	9, 5	resubcld 8527	. . . . 5 |- ( ph -> ( 1 - A ) e. RR )
30	5, 9	posdifd 8679	. . . . . 6 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
31	11, 30	mpbid 147	. . . . 5 |- ( ph -> 0 < ( 1 - A ) )
32	29, 31	elrpd 9889	. . . 4 |- ( ph -> ( 1 - A ) e. RR+ )
33	7, 32	rpdivcld 9910	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
34	28, 33	rpmulcld 9909	. 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 12041	. . 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 2612	. 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 11861	1 |- ( ph -> seq 1 ( + , F ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083   dom cdm 4719   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   <_ cle 8182   - cmin 8317   / cdiv 8819   NNcn 9110   ZZ>=cuz 9722   seqcseq 10669   abscabs 11508   ~~> cli 11789

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ico 10090  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  cvgratz  12043