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Theorem trirecip 13731
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 10567 . . . 4  |-  ( k  e.  NN  ->  2  e.  CC )
2 peano2nn 10506 . . . . . 6  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3 nnmulcl 10517 . . . . . 6  |-  ( ( k  e.  NN  /\  ( k  +  1 )  e.  NN )  ->  ( k  x.  ( k  +  1 ) )  e.  NN )
42, 3mpdan 666 . . . . 5  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
54nncnd 10510 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  CC )
64nnne0d 10539 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  =/=  0 )
71, 5, 6divrecd 10282 . . 3  |-  ( k  e.  NN  ->  (
2  /  ( k  x.  ( k  +  1 ) ) )  =  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
87sumeq2i 13575 . 2  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  sum_ k  e.  NN  (
2  x.  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
9 nnuz 11078 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
10 1zzd 10854 . . . . 5  |-  ( T. 
->  1  e.  ZZ )
11 id 22 . . . . . . . . 9  |-  ( n  =  k  ->  n  =  k )
12 oveq1 6239 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1311, 12oveq12d 6250 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  ( n  +  1 ) )  =  ( k  x.  ( k  +  1 ) ) )
1413oveq2d 6248 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  ( n  x.  ( n  + 
1 ) ) )  =  ( 1  / 
( k  x.  (
k  +  1 ) ) ) )
15 eqid 2400 . . . . . . 7  |-  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) )
16 ovex 6260 . . . . . . 7  |-  ( 1  /  ( k  x.  ( k  +  1 ) ) )  e. 
_V
1714, 15, 16fvmpt 5886 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
1817adantl 464 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
194nnrecred 10540 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
2019recnd 9570 . . . . . 6  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2120adantl 464 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2215trireciplem 13730 . . . . . . 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 13369 . . . . . . 7  |-  Rel  ~~>
2524releldmi 5179 . . . . . 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 17 . . . . 5  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  e.  dom  ~~>  )
27 2cnd 10567 . . . . 5  |-  ( T. 
->  2  e.  CC )
289, 10, 18, 21, 26, 27isummulc2 13633 . . . 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 13628 . . . . 5  |-  ( T. 
->  sum_ k  e.  NN  ( 1  /  (
k  x.  ( k  +  1 ) ) )  =  1 )
3029oveq2d 6248 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3128, 30eqtr3d 2443 . . 3  |-  ( T. 
->  sum_ k  e.  NN  ( 2  x.  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3231trud 1412 . 2  |-  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 )
33 2t1e2 10643 . 2  |-  ( 2  x.  1 )  =  2
348, 32, 333eqtri 2433 1  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403   T. wtru 1404    e. wcel 1840   class class class wbr 4392    |-> cmpt 4450   dom cdm 4940   ` cfv 5523  (class class class)co 6232   CCcc 9438   1c1 9441    + caddc 9443    x. cmul 9445    / cdiv 10165   NNcn 10494   2c2 10544    seqcseq 12059    ~~> cli 13361   sum_csu 13562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fz 11642  df-fzo 11766  df-fl 11877  df-seq 12060  df-exp 12119  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-rlim 13366  df-sum 13563
This theorem is referenced by: (None)
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