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Theorem cvgratnn 10912
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 10913 and cvgratgt0 10914, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 10726 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.
StepHypRef Expression
1 nnuz 9044 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 8767 . . 3  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 9888 . 2  |-  ( ph  ->  seq 1 (  +  ,  F ) : NN --> CC )
5 cvgratnn.3 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
6 cvgratnn.gt0 . . . . . . . . . 10  |-  ( ph  ->  0  <  A )
75, 6elrpd 9161 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
87rprecred 9175 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
9 1red 7493 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
108, 9resubcld 7849 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR )
11 cvgratnn.4 . . . . . . . . 9  |-  ( ph  ->  A  <  1 )
127reclt1d 9177 . . . . . . . . 9  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
1311, 12mpbid 145 . . . . . . . 8  |-  ( ph  ->  1  <  ( 1  /  A ) )
149, 8posdifd 7999 . . . . . . . 8  |-  ( ph  ->  ( 1  <  (
1  /  A )  <->  0  <  ( ( 1  /  A )  -  1 ) ) )
1513, 14mpbid 145 . . . . . . 7  |-  ( ph  ->  0  <  ( ( 1  /  A )  -  1 ) )
1610, 15elrpd 9161 . . . . . 6  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
1716rpreccld 9174 . . . . 5  |-  ( ph  ->  ( 1  /  (
( 1  /  A
)  -  1 ) )  e.  RR+ )
1817, 7rpdivcld 9181 . . . 4  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  e.  RR+ )
19 fveq2 5299 . . . . . . . 8  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
2019eleq1d 2156 . . . . . . 7  |-  ( k  =  1  ->  (
( F `  k
)  e.  CC  <->  ( F `  1 )  e.  CC ) )
213ralrimiva 2446 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
22 1nn 8423 . . . . . . . 8  |-  1  e.  NN
2322a1i 9 . . . . . . 7  |-  ( ph  ->  1  e.  NN )
2420, 21, 23rspcdva 2727 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  CC )
2524abscld 10602 . . . . 5  |-  ( ph  ->  ( abs `  ( F `  1 )
)  e.  RR )
2624absge0d 10605 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  ( F `  1
) ) )
2725, 26ge0p1rpd 9194 . . . 4  |-  ( ph  ->  ( ( abs `  ( F `  1 )
)  +  1 )  e.  RR+ )
2818, 27rpmulcld 9180 . . 3  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR+ )
299, 5resubcld 7849 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
305, 9posdifd 7999 . . . . . 6  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
3111, 30mpbid 145 . . . . 5  |-  ( ph  ->  0  <  ( 1  -  A ) )
3229, 31elrpd 9161 . . . 4  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
337, 32rpdivcld 9181 . . 3  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR+ )
3428, 33rpmulcld 9180 . 2  |-  ( ph  ->  ( ( ( ( 1  /  ( ( 1  /  A )  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1 )
)  +  1 ) )  x.  ( A  /  ( 1  -  A ) ) )  e.  RR+ )
355adantr 270 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  n  e.  ( ZZ>= `  m )
) )  ->  A  e.  RR )
3611adantr 270 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  n  e.  ( ZZ>= `  m )
) )  ->  A  <  1 )
376adantr 270 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  n  e.  ( ZZ>= `  m )
) )  ->  0  <  A )
383adantlr 461 . . . 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
) ) ) )
4039adantlr 461 . . . 4  |-  ( ( ( ph  /\  (
m  e.  NN  /\  n  e.  ( ZZ>= `  m ) ) )  /\  k  e.  NN )  ->  ( abs `  ( F `  ( k  +  1 ) ) )  <_  ( A  x.  ( abs `  ( F `  k )
) ) )
41 simprl 498 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  n  e.  ( ZZ>= `  m )
) )  ->  m  e.  NN )
42 simprr 499 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  n  e.  ( ZZ>= `  m )
) )  ->  n  e.  ( ZZ>= `  m )
)
4335, 36, 37, 38, 40, 41, 42cvgratnnlemrate 10911 . . 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 ) )
4443ralrimivva 2455 . 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 ) )
454, 34, 44climcvg1n 10726 1  |-  ( ph  ->  seq 1 (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   class class class wbr 3843   dom cdm 4436   ` cfv 5010  (class class class)co 5644   CCcc 7338   RRcr 7339   0cc0 7340   1c1 7341    + caddc 7343    x. cmul 7345    < clt 7512    <_ cle 7513    - cmin 7643    / cdiv 8129   NNcn 8412   ZZ>=cuz 9009    seqcseq 9840   abscabs 10418    ~~> cli 10653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453  ax-arch 7454  ax-caucvg 7455
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-isom 5019  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-frec 6148  df-1o 6173  df-oadd 6177  df-er 6282  df-en 6448  df-dom 6449  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-inn 8413  df-2 8471  df-3 8472  df-4 8473  df-n0 8664  df-z 8741  df-uz 9010  df-q 9095  df-rp 9125  df-ico 9302  df-fz 9415  df-fzo 9542  df-iseq 9841  df-seq3 9842  df-exp 9943  df-ihash 10172  df-cj 10264  df-re 10265  df-im 10266  df-rsqrt 10419  df-abs 10420  df-clim 10654  df-isum 10730
This theorem is referenced by:  cvgratz  10913
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