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Theorem geolim 12685
Description: The partial sums in the infinite series  1  +  A ^ 1  +  A ^ 2... converge to  ( 1  /  (
1  -  A ) ). (Contributed by NM, 15-May-2006.)
Hypotheses
Ref Expression
geolim.1  |-  ( ph  ->  A  e.  CC )
geolim.2  |-  ( ph  ->  ( abs `  A
)  <  1 )
geolim.3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( A ^ k ) )
Assertion
Ref Expression
geolim  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Distinct variable groups:    A, k    k, F    ph, k

Proof of Theorem geolim
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10558 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10331 . . . 4  |-  0  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
4 geolim.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
5 geolim.2 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  <  1 )
64, 5expcnv 12681 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
7 ax-1cn 9086 . . . . . . 7  |-  1  e.  CC
8 subcl 9343 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
97, 4, 8sylancr 646 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
10 1re 9128 . . . . . . . . . . . 12  |-  1  e.  RR
1110ltnri 9220 . . . . . . . . . . 11  |-  -.  1  <  1
12 fveq2 5763 . . . . . . . . . . . . 13  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
13 abs1 12140 . . . . . . . . . . . . 13  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2491 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
1514breq1d 4253 . . . . . . . . . . 11  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
1611, 15mtbiri 296 . . . . . . . . . 10  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
1716necon2ai 2656 . . . . . . . . 9  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
185, 17syl 16 . . . . . . . 8  |-  ( ph  ->  A  =/=  1 )
1918necomd 2694 . . . . . . 7  |-  ( ph  ->  1  =/=  A )
20 subeq0 9365 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
217, 4, 20sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
2221necon3bid 2643 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
2319, 22mpbird 225 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
244, 9, 23divcld 9828 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
25 nn0ex 10265 . . . . . . 7  |-  NN0  e.  _V
2625mptex 6002 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  e.  _V
2726a1i 11 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  e.  _V )
28 oveq2 6125 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
29 eqid 2443 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
30 ovex 6142 . . . . . . . 8  |-  ( A ^ j )  e. 
_V
3128, 29, 30fvmpt 5842 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
3231adantl 454 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
33 expcl 11437 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
344, 33sylan 459 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
3532, 34eqeltrd 2517 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
36 expp1 11426 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ (
j  +  1 ) )  =  ( ( A ^ j )  x.  A ) )
374, 36sylan 459 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( ( A ^
j )  x.  A
) )
384adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
3934, 38mulcomd 9147 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
4037, 39eqtrd 2475 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
4140oveq1d 6132 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) ) )
429adantr 453 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  e.  CC )
4323adantr 453 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  =/=  0 )
4438, 34, 42, 43div23d 9865 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
4541, 44eqtrd 2475 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
46 oveq1 6124 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4746oveq2d 6133 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4847oveq1d 6132 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
49 eqid 2443 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
50 ovex 6142 . . . . . . . 8  |-  ( ( A ^ ( j  +  1 ) )  /  ( 1  -  A ) )  e. 
_V
5148, 49, 50fvmpt 5842 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5251adantl 454 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5332oveq2d 6133 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5445, 52, 533eqtr4d 2485 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A  / 
( 1  -  A
) )  x.  (
( n  e.  NN0  |->  ( A ^ n ) ) `  j ) ) )
551, 3, 6, 24, 27, 35, 54climmulc2 12468 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5624mul01d 9303 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5755, 56breqtrd 4267 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
589, 23reccld 9821 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
59 seqex 11363 . . . 4  |-  seq  0
(  +  ,  F
)  e.  _V
6059a1i 11 . . 3  |-  ( ph  ->  seq  0 (  +  ,  F )  e. 
_V )
61 peano2nn0 10298 . . . . . 6  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
62 expcl 11437 . . . . . 6  |-  ( ( A  e.  CC  /\  ( j  +  1 )  e.  NN0 )  ->  ( A ^ (
j  +  1 ) )  e.  CC )
634, 61, 62syl2an 465 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
6463, 42, 43divcld 9828 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6552, 64eqeltrd 2517 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
66 nn0cn 10269 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6766adantl 454 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
68 pncan 9349 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6967, 7, 68sylancl 645 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
7069oveq2d 6133 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
7170sumeq1d 12533 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  sum_ k  e.  ( 0 ... j ) ( A ^ k ) )
727a1i 11 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  1  e.  CC )
7372, 63, 42, 43divsubdird 9867 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) )  =  ( ( 1  / 
( 1  -  A
) )  -  (
( A ^ (
j  +  1 ) )  /  ( 1  -  A ) ) ) )
7418adantr 453 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  =/=  1 )
7561adantl 454 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
7638, 74, 75geoser 12684 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7752oveq2d 6133 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  /  ( 1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) ) )
7873, 76, 773eqtr4d 2485 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  /  (
1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) ) )
79 simpll 732 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ph )
80 elfznn0 11121 . . . . . . 7  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
8180adantl 454 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  k  e.  NN0 )
82 geolim.3 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( A ^ k ) )
8379, 81, 82syl2anc 644 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( F `  k )  =  ( A ^
k ) )
84 simpr 449 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
8584, 1syl6eleq 2533 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  ( ZZ>= `  0 )
)
8679, 4syl 16 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  A  e.  CC )
8786, 81expcld 11561 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( A ^ k )  e.  CC )
8883, 85, 87fsumser 12562 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq  0 (  +  ,  F ) `  j ) )
8971, 78, 883eqtr3rd 2484 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  (  seq  0 (  +  ,  F ) `  j
)  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) ) )
901, 3, 57, 58, 60, 65, 89climsubc2 12470 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
9158subid1d 9438 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
9290, 91breqtrd 4267 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   _Vcvv 2965   class class class wbr 4243    e. cmpt 4297   ` cfv 5489  (class class class)co 6117   CCcc 9026   0cc0 9028   1c1 9029    + caddc 9031    x. cmul 9033    < clt 9158    - cmin 9329    / cdiv 9715   NN0cn0 10259   ZZcz 10320   ZZ>=cuz 10526   ...cfz 11081    seq cseq 11361   ^cexp 11420   abscabs 12077    ~~> cli 12316   sum_csu 12517
This theorem is referenced by:  geolim2  12686  georeclim  12687  geomulcvg  12691  geoisum  12692  cvgrat  12698  eflegeo  12760  geolim3  20294  abelthlem5  20389  logtayllem  20588  zetacvg  24834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-inf2 7632  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105  ax-pre-sup 9106
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-se 4577  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-isom 5498  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-pm 7057  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-sup 7482  df-oi 7515  df-card 7864  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-nn 10039  df-2 10096  df-3 10097  df-n0 10260  df-z 10321  df-uz 10527  df-rp 10651  df-fz 11082  df-fzo 11174  df-fl 11240  df-seq 11362  df-exp 11421  df-hash 11657  df-cj 11942  df-re 11943  df-im 11944  df-sqr 12078  df-abs 12079  df-clim 12320  df-rlim 12321  df-sum 12518
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