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Theorem geoisumr 11553
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 9587 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9290 . 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 11217 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( abs `  A
)  e.  RR )
6 0red 7983 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
0  e.  RR )
7 1red 7997 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
1  e.  RR )
8 0lt1 8109 . . . . . . . . 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 8108 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
0  <  ( abs `  A ) )
125, 11gt0ap0d 8611 . . . . . 6  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( abs `  A
) #  0 )
13 abs00ap 11098 . . . . . . 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 8763 . . . 4  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( 1  /  A
)  e.  CC )
1716, 3expcld 10680 . . 3  |-  ( ( ( A  e.  CC  /\  1  <  ( abs `  A ) )  /\  k  e.  NN0 )  -> 
( ( 1  /  A ) ^ k
)  e.  CC )
18 oveq2 5900 . . . 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 5609 . . 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 11548 . 2  |-  ( ( A  e.  CC  /\  1  <  ( abs `  A
) )  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( 1  /  A ) ^ n
) ) )  ~~>  ( A  /  ( A  - 
1 ) ) )
251, 2, 21, 17, 24isumclim 11456 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 5232  (class class class)co 5892   CCcc 7834   0cc0 7836   1c1 7837    < clt 8017    - cmin 8153   # cap 8563    / cdiv 8654   NN0cn0 9201   ^cexp 10545   abscabs 11033   sum_csu 11388
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 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955  ax-caucvg 7956
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 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-frec 6411  df-1o 6436  df-oadd 6440  df-er 6554  df-en 6762  df-dom 6763  df-fin 6764  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-3 9004  df-4 9005  df-n0 9202  df-z 9279  df-uz 9554  df-q 9645  df-rp 9679  df-fz 10034  df-fzo 10168  df-seqfrec 10472  df-exp 10546  df-ihash 10783  df-cj 10878  df-re 10879  df-im 10880  df-rsqrt 11034  df-abs 11035  df-clim 11314  df-sumdc 11389
This theorem is referenced by: (None)
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