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Theorem trirecip 11476
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 8963 . . . 4  |-  ( k  e.  NN  ->  2  e.  CC )
2 peano2nn 8902 . . . . . 6  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3 nnmulcl 8911 . . . . . 6  |-  ( ( k  e.  NN  /\  ( k  +  1 )  e.  NN )  ->  ( k  x.  ( k  +  1 ) )  e.  NN )
42, 3mpdan 421 . . . . 5  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
54nncnd 8904 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) )  e.  CC )
64nnap0d 8936 . . . 4  |-  ( k  e.  NN  ->  (
k  x.  ( k  +  1 ) ) #  0 )
71, 5, 6divrecapd 8722 . . 3  |-  ( k  e.  NN  ->  (
2  /  ( k  x.  ( k  +  1 ) ) )  =  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) ) )
87sumeq2i 11339 . 2  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  sum_ k  e.  NN  (
2  x.  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
9 nnuz 9534 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
10 1zzd 9251 . . . . 5  |-  ( T. 
->  1  e.  ZZ )
11 simpr 110 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  k  e.  NN )
124adantl 277 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
k  x.  ( k  +  1 ) )  e.  NN )
1312nnrecred 8937 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
14 id 19 . . . . . . . . 9  |-  ( n  =  k  ->  n  =  k )
15 oveq1 5872 . . . . . . . . 9  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
1614, 15oveq12d 5883 . . . . . . . 8  |-  ( n  =  k  ->  (
n  x.  ( n  +  1 ) )  =  ( k  x.  ( k  +  1 ) ) )
1716oveq2d 5881 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  ( n  x.  ( n  + 
1 ) ) )  =  ( 1  / 
( k  x.  (
k  +  1 ) ) ) )
18 eqid 2175 . . . . . . 7  |-  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 ) ) ) )  =  ( n  e.  NN  |->  ( 1  / 
( n  x.  (
n  +  1 ) ) ) )
1917, 18fvmptg 5584 . . . . . 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 411 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( 1  /  (
n  x.  ( n  +  1 ) ) ) ) `  k
)  =  ( 1  /  ( k  x.  ( k  +  1 ) ) ) )
214nnrecred 8937 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  RR )
2221recnd 7960 . . . . . 6  |-  ( k  e.  NN  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2322adantl 277 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  /  ( k  x.  ( k  +  1 ) ) )  e.  CC )
2418trireciplem 11475 . . . . . . 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 11255 . . . . . . 7  |-  Rel  ~~>
2726releldmi 4859 . . . . . 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 8963 . . . . 5  |-  ( T. 
->  2  e.  CC )
309, 10, 20, 23, 28, 29isummulc2 11401 . . . 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 11396 . . . . 5  |-  ( T. 
->  sum_ k  e.  NN  ( 1  /  (
k  x.  ( k  +  1 ) ) )  =  1 )
3231oveq2d 5881 . . . 4  |-  ( T. 
->  ( 2  x.  sum_ k  e.  NN  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3330, 32eqtr3d 2210 . . 3  |-  ( T. 
->  sum_ k  e.  NN  ( 2  x.  (
1  /  ( k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 ) )
3433mptru 1362 . 2  |-  sum_ k  e.  NN  ( 2  x.  ( 1  /  (
k  x.  ( k  +  1 ) ) ) )  =  ( 2  x.  1 )
35 2t1e2 9043 . 2  |-  ( 2  x.  1 )  =  2
368, 34, 353eqtri 2200 1  |-  sum_ k  e.  NN  ( 2  / 
( k  x.  (
k  +  1 ) ) )  =  2
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   T. wtru 1354    e. wcel 2146   class class class wbr 3998    |-> cmpt 4059   dom cdm 4620   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   1c1 7787    + caddc 7789    x. cmul 7791    / cdiv 8601   NNcn 8890   2c2 8941    seqcseq 10413    ~~> cli 11253   sum_csu 11328
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-exp 10488  df-ihash 10722  df-shft 10791  df-cj 10818  df-re 10819  df-im 10820  df-rsqrt 10974  df-abs 10975  df-clim 11254  df-sumdc 11329
This theorem is referenced by: (None)
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