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Theorem trirecip 13628
Description: The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
Assertion
Ref Expression
trirecip  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2

Proof of Theorem trirecip
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 2cnd 10599 . . . 4  |-  ( k  e.  NN  ->  2  e.  CC )
2 peano2nn 10539 . . . . . 6  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3 nnmulcl 10550 . . . . . 6  |-  ( ( k  e.  NN  /\  ( k  +  1 )  e.  NN )  ->  ( k  x.  ( k  +  1 ) )  e.  NN )
42, 3mpdan 668 . . . . 5  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
54nncnd 10543 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  CC )
64nnne0d 10571 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  =/=  0 )
71, 5, 6divrecd 10314 . . 3  |-  ( k  e.  NN  ->  (
2  /  ( k  x.  ( k  +  1 ) ) )  =  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
87sumeq2i 13472 . 2  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  sum_ k  e.  NN  (
2  x.  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
9 nnuz 11108 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
10 1zzd 10886 . . . . 5  |-  ( T. 
->  1  e.  ZZ )
11 id 22 . . . . . . . . 9  |-  ( n  =  k  ->  n  =  k )
12 oveq1 6284 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1311, 12oveq12d 6295 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  ( n  +  1 ) )  =  ( k  x.  ( k  +  1 ) ) )
1413oveq2d 6293 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  ( n  x.  ( n  + 
1 ) ) )  =  ( 1  / 
( k  x.  (
k  +  1 ) ) ) )
15 eqid 2462 . . . . . . 7  |-  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) )
16 ovex 6302 . . . . . . 7  |-  ( 1  /  ( k  x.  ( k  +  1 ) ) )  e. 
_V
1714, 15, 16fvmpt 5943 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
1817adantl 466 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
194nnrecred 10572 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
2019recnd 9613 . . . . . 6  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2120adantl 466 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2215trireciplem 13627 . . . . . . 7  |-  seq 1
(  +  ,  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  + 
1 ) ) ) ) )  ~~>  1
2322a1i 11 . . . . . 6  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  ~~>  1 )
24 climrel 13266 . . . . . . 7  |-  Rel  ~~>
2524releldmi 5232 . . . . . 6  |-  (  seq 1 (  +  , 
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) )  ~~>  1  ->  seq 1 (  +  , 
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) )  e. 
dom 
~~>  )
2623, 25syl 16 . . . . 5  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  e.  dom  ~~>  )
27 2cnd 10599 . . . . 5  |-  ( T. 
->  2  e.  CC )
289, 10, 18, 21, 26, 27isummulc2 13528 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
299, 10, 18, 21, 23isumclim 13523 . . . . 5  |-  ( T. 
->  sum_ k  e.  NN  ( 1  /  (
k  x.  ( k  +  1 ) ) )  =  1 )
3029oveq2d 6293 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3128, 30eqtr3d 2505 . . 3  |-  ( T. 
->  sum_ k  e.  NN  ( 2  x.  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3231trud 1383 . 2  |-  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 )
33 2t1e2 10675 . 2  |-  ( 2  x.  1 )  =  2
348, 32, 333eqtri 2495 1  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   T. wtru 1375    e. wcel 1762   class class class wbr 4442    |-> cmpt 4500   dom cdm 4994   ` cfv 5581  (class class class)co 6277   CCcc 9481   1c1 9484    + caddc 9486    x. cmul 9488    / cdiv 10197   NNcn 10527   2c2 10576    seqcseq 12065    ~~> cli 13258   sum_csu 13459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460
This theorem is referenced by: (None)
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