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Theorem geoisumr 11545
Description: The infinite sum of reciprocals  1  +  ( 1  /  A ) ^ 1  +  ( 1  /  A ) ^ 2... is  A  / 
( A  -  1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
Assertion
Ref Expression
geoisumr  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  sum_ k  e.  NN0  ( ( 1  /  A ) ^
k )  =  ( A  /  ( A  -  1 ) ) )
Distinct variable group:    A, k

Proof of Theorem geoisumr
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 nn0uz 9581 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9284 . 2  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  0  e.  ZZ )
3 simpr 110 . . 3  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
k  e.  NN0 )
4 simpll 527 . . . . 5  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  ->  A  e.  CC )
54abscld 11209 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( abs `  A
)  e.  RR )
6 0red 7977 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
0  e.  RR )
7 1red 7991 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
1  e.  RR )
8 0lt1 8103 . . . . . . . . 9  |-  0  <  1
98a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
0  <  1 )
10 simplr 528 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
1  <  ( abs `  A ) )
116, 7, 5, 9, 10lttrd 8102 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
0  <  ( abs `  A ) )
125, 11gt0ap0d 8605 . . . . . 6  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( abs `  A
) #  0 )
13 abs00ap 11090 . . . . . . 7  |-  ( A  e.  CC  ->  (
( abs `  A
) #  0  <->  A #  0
) )
1413ad2antrr 488 . . . . . 6  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( ( abs `  A
) #  0  <->  A #  0
) )
1512, 14mpbid 147 . . . . 5  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  ->  A #  0 )
164, 15recclapd 8757 . . . 4  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( 1  /  A
)  e.  CC )
1716, 3expcld 10673 . . 3  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( ( 1  /  A ) ^ k
)  e.  CC )
18 oveq2 5899 . . . 4  |-  ( n  =  k  ->  (
( 1  /  A
) ^ n )  =  ( ( 1  /  A ) ^
k ) )
19 eqid 2189 . . . 4  |-  ( n  e.  NN0  |->  ( ( 1  /  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( 1  /  A ) ^
n ) )
2018, 19fvmptg 5608 . . 3  |-  ( ( k  e.  NN0  /\  ( ( 1  /  A ) ^ k
)  e.  CC )  ->  ( ( n  e.  NN0  |->  ( ( 1  /  A ) ^ n ) ) `
 k )  =  ( ( 1  /  A ) ^ k
) )
213, 17, 20syl2anc 411 . 2  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( 1  /  A ) ^
n ) ) `  k )  =  ( ( 1  /  A
) ^ k ) )
22 simpl 109 . . 3  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  A  e.  CC )
23 simpr 110 . . 3  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  1  <  ( abs `  A
) )
2422, 23, 21georeclim 11540 . 2  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( 1  /  A ) ^ n
) ) )  ~~>  ( A  /  ( A  - 
1 ) ) )
251, 2, 21, 17, 24isumclim 11448 1  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  sum_ k  e.  NN0  ( ( 1  /  A ) ^
k )  =  ( A  /  ( A  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   class class class wbr 4018    |-> cmpt 4079   ` cfv 5231  (class class class)co 5891   CCcc 7828   0cc0 7830   1c1 7831    < clt 8011    - cmin 8147   # cap 8557    / cdiv 8648   NN0cn0 9195   ^cexp 10538   abscabs 11025   sum_csu 11380
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-fz 10028  df-fzo 10162  df-seqfrec 10465  df-exp 10539  df-ihash 10775  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-clim 11306  df-sumdc 11381
This theorem is referenced by: (None)
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