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Theorem trirecip 11282
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 8805 . . . 4  |-  ( k  e.  NN  ->  2  e.  CC )
2 peano2nn 8744 . . . . . 6  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3 nnmulcl 8753 . . . . . 6  |-  ( ( k  e.  NN  /\  ( k  +  1 )  e.  NN )  ->  ( k  x.  ( k  +  1 ) )  e.  NN )
42, 3mpdan 417 . . . . 5  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
54nncnd 8746 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  CC )
64nnap0d 8778 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) ) #  0 )
71, 5, 6divrecapd 8565 . . 3  |-  ( k  e.  NN  ->  (
2  /  ( k  x.  ( k  +  1 ) ) )  =  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
87sumeq2i 11145 . 2  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  sum_ k  e.  NN  (
2  x.  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
9 nnuz 9373 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
10 1zzd 9093 . . . . 5  |-  ( T. 
->  1  e.  ZZ )
11 simpr 109 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  k  e.  NN )
124adantl 275 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
1312nnrecred 8779 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
14 id 19 . . . . . . . . 9  |-  ( n  =  k  ->  n  =  k )
15 oveq1 5781 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1614, 15oveq12d 5792 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  ( n  +  1 ) )  =  ( k  x.  ( k  +  1 ) ) )
1716oveq2d 5790 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  ( n  x.  ( n  + 
1 ) ) )  =  ( 1  / 
( k  x.  (
k  +  1 ) ) ) )
18 eqid 2139 . . . . . . 7  |-  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) )
1917, 18fvmptg 5497 . . . . . 6  |-  ( ( k  e.  NN  /\  ( 1  /  (
k  x.  ( k  +  1 ) ) )  e.  RR )  ->  ( ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) ) `
 k )  =  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )
2011, 13, 19syl2anc 408 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
214nnrecred 8779 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
2221recnd 7806 . . . . . 6  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2322adantl 275 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2418trireciplem 11281 . . . . . . 7  |-  seq 1
(  +  ,  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  + 
1 ) ) ) ) )  ~~>  1
2524a1i 9 . . . . . 6  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  ~~>  1 )
26 climrel 11061 . . . . . . 7  |-  Rel  ~~>
2726releldmi 4778 . . . . . 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 14 . . . . 5  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) ) )  e.  dom  ~~>  )
29 2cnd 8805 . . . . 5  |-  ( T. 
->  2  e.  CC )
309, 10, 20, 23, 28, 29isummulc2 11207 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
319, 10, 20, 23, 25isumclim 11202 . . . . 5  |-  ( T. 
->  sum_ k  e.  NN  ( 1  /  (
k  x.  ( k  +  1 ) ) )  =  1 )
3231oveq2d 5790 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3330, 32eqtr3d 2174 . . 3  |-  ( T. 
->  sum_ k  e.  NN  ( 2  x.  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3433mptru 1340 . 2  |-  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 )
35 2t1e2 8885 . 2  |-  ( 2  x.  1 )  =  2
368, 34, 353eqtri 2164 1  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   T. wtru 1332    e. wcel 1480   class class class wbr 3929    |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   1c1 7633    + caddc 7635    x. cmul 7637    / cdiv 8444   NNcn 8732   2c2 8783    seqcseq 10230    ~~> cli 11059   sum_csu 11134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-ihash 10534  df-shft 10599  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135
This theorem is referenced by: (None)
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