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Theorem geolim 12610
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 10484 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10257 . . . 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 12606 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
7 ax-1cn 9012 . . . . . . 7  |-  1  e.  CC
8 subcl 9269 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
97, 4, 8sylancr 645 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
10 1re 9054 . . . . . . . . . . . 12  |-  1  e.  RR
1110ltnri 9146 . . . . . . . . . . 11  |-  -.  1  <  1
12 fveq2 5695 . . . . . . . . . . . . 13  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
13 abs1 12065 . . . . . . . . . . . . 13  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2460 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
1514breq1d 4190 . . . . . . . . . . 11  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
1611, 15mtbiri 295 . . . . . . . . . 10  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
1716necon2ai 2620 . . . . . . . . 9  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
185, 17syl 16 . . . . . . . 8  |-  ( ph  ->  A  =/=  1 )
1918necomd 2658 . . . . . . 7  |-  ( ph  ->  1  =/=  A )
20 subeq0 9291 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
217, 4, 20sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
2221necon3bid 2610 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
2319, 22mpbird 224 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
244, 9, 23divcld 9754 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
25 nn0ex 10191 . . . . . . 7  |-  NN0  e.  _V
2625mptex 5933 . . . . . 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 6056 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
29 eqid 2412 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
30 ovex 6073 . . . . . . . 8  |-  ( A ^ j )  e. 
_V
3128, 29, 30fvmpt 5773 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
3231adantl 453 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
33 expcl 11362 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
344, 33sylan 458 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
3532, 34eqeltrd 2486 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
36 expp1 11351 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ (
j  +  1 ) )  =  ( ( A ^ j )  x.  A ) )
374, 36sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( ( A ^
j )  x.  A
) )
384adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
3934, 38mulcomd 9073 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
4037, 39eqtrd 2444 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
4140oveq1d 6063 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) ) )
429adantr 452 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  e.  CC )
4323adantr 452 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  =/=  0 )
4438, 34, 42, 43div23d 9791 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
4541, 44eqtrd 2444 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
46 oveq1 6055 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4746oveq2d 6064 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4847oveq1d 6063 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
49 eqid 2412 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
50 ovex 6073 . . . . . . . 8  |-  ( ( A ^ ( j  +  1 ) )  /  ( 1  -  A ) )  e. 
_V
5148, 49, 50fvmpt 5773 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5251adantl 453 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5332oveq2d 6064 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5445, 52, 533eqtr4d 2454 . . . . 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 12393 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5624mul01d 9229 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5755, 56breqtrd 4204 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
589, 23reccld 9747 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
59 seqex 11288 . . . 4  |-  seq  0
(  +  ,  F
)  e.  _V
6059a1i 11 . . 3  |-  ( ph  ->  seq  0 (  +  ,  F )  e. 
_V )
61 peano2nn0 10224 . . . . . 6  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
62 expcl 11362 . . . . . 6  |-  ( ( A  e.  CC  /\  ( j  +  1 )  e.  NN0 )  ->  ( A ^ (
j  +  1 ) )  e.  CC )
634, 61, 62syl2an 464 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
6463, 42, 43divcld 9754 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6552, 64eqeltrd 2486 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
66 nn0cn 10195 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6766adantl 453 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
68 pncan 9275 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6967, 7, 68sylancl 644 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
7069oveq2d 6064 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
7170sumeq1d 12458 . . . 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 9793 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) )  =  ( ( 1  / 
( 1  -  A
) )  -  (
( A ^ (
j  +  1 ) )  /  ( 1  -  A ) ) ) )
7418adantr 452 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  =/=  1 )
7561adantl 453 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
7638, 74, 75geoser 12609 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7752oveq2d 6064 . . . . 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 2454 . . . 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 731 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ph )
80 elfznn0 11047 . . . . . . 7  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
8180adantl 453 . . . . . 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 643 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( F `  k )  =  ( A ^
k ) )
84 simpr 448 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
8584, 1syl6eleq 2502 . . . . 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 11486 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( A ^ k )  e.  CC )
8883, 85, 87fsumser 12487 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq  0 (  +  ,  F ) `  j ) )
8971, 78, 883eqtr3rd 2453 . . 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 12395 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
9158subid1d 9364 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
9290, 91breqtrd 4204 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    - cmin 9255    / cdiv 9641   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007    seq cseq 11286   ^cexp 11345   abscabs 12002    ~~> cli 12241   sum_csu 12442
This theorem is referenced by:  geolim2  12611  georeclim  12612  geomulcvg  12616  geoisum  12617  cvgrat  12623  eflegeo  12685  geolim3  20217  abelthlem5  20312  logtayllem  20511  zetacvg  24760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443
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