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Theorem trirecip 12368
Description: The sum of the reciprocals of the triangle numbers converge to two. (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 2cn 9861 . . . . 5  |-  2  e.  CC
21a1i 10 . . . 4  |-  ( k  e.  NN  ->  2  e.  CC )
3 peano2nn 9803 . . . . . 6  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
4 nnmulcl 9814 . . . . . 6  |-  ( ( k  e.  NN  /\  ( k  +  1 )  e.  NN )  ->  ( k  x.  ( k  +  1 ) )  e.  NN )
53, 4mpdan 649 . . . . 5  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
65nncnd 9807 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  CC )
75nnne0d 9835 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  =/=  0 )
82, 6, 7divrecd 9584 . . 3  |-  ( k  e.  NN  ->  (
2  /  ( k  x.  ( k  +  1 ) ) )  =  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
98sumeq2i 12219 . 2  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  sum_ k  e.  NN  (
2  x.  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
10 nnuz 10310 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
11 1z 10100 . . . . . 6  |-  1  e.  ZZ
1211a1i 10 . . . . 5  |-  (  T. 
->  1  e.  ZZ )
13 id 19 . . . . . . . . 9  |-  ( n  =  k  ->  n  =  k )
14 oveq1 5907 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1513, 14oveq12d 5918 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  ( n  +  1 ) )  =  ( k  x.  ( k  +  1 ) ) )
1615oveq2d 5916 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  ( n  x.  ( n  + 
1 ) ) )  =  ( 1  / 
( k  x.  (
k  +  1 ) ) ) )
17 eqid 2316 . . . . . . 7  |-  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) )
18 ovex 5925 . . . . . . 7  |-  ( 1  /  ( k  x.  ( k  +  1 ) ) )  e. 
_V
1916, 17, 18fvmpt 5640 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
2019adantl 452 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
215nnrecred 9836 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
2221recnd 8906 . . . . . 6  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2322adantl 452 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2417trireciplem 12367 . . . . . . 7  |-  seq  1
(  +  ,  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  + 
1 ) ) ) ) )  ~~>  1
2524a1i 10 . . . . . 6  |-  (  T. 
->  seq  1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  ~~>  1 )
26 climrel 12013 . . . . . . 7  |-  Rel  ~~>
2726releldmi 4952 . . . . . 6  |-  (  seq  1 (  +  , 
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) )  ~~>  1  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) )  e. 
dom 
~~>  )
2825, 27syl 15 . . . . 5  |-  (  T. 
->  seq  1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  e.  dom  ~~>  )
291a1i 10 . . . . 5  |-  (  T. 
->  2  e.  CC )
3010, 12, 20, 23, 28, 29isummulc2 12272 . . . 4  |-  (  T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
3110, 12, 20, 23, 25isumclim 12267 . . . . 5  |-  (  T. 
->  sum_ k  e.  NN  ( 1  /  (
k  x.  ( k  +  1 ) ) )  =  1 )
3231oveq2d 5916 . . . 4  |-  (  T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3330, 32eqtr3d 2350 . . 3  |-  (  T. 
->  sum_ k  e.  NN  ( 2  x.  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3433trud 1314 . 2  |-  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 )
351mulid1i 8884 . 2  |-  ( 2  x.  1 )  =  2
369, 34, 353eqtri 2340 1  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2
Colors of variables: wff set class
Syntax hints:    T. wtru 1307    = wceq 1633    e. wcel 1701   class class class wbr 4060    e. cmpt 4114   dom cdm 4726   ` cfv 5292  (class class class)co 5900   CCcc 8780   1c1 8783    + caddc 8785    x. cmul 8787    / cdiv 9468   NNcn 9791   2c2 9840   ZZcz 10071    seq cseq 11093    ~~> cli 12005   sum_csu 12205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-rlim 12010  df-sum 12206
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