Theorem trirecip 11442

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.

Step	Hyp	Ref	Expression
1		2cnd 8930	. . . 4 |- ( k e. NN -> 2 e. CC )
2		peano2nn 8869	. . . . . 6 |- ( k e. NN -> ( k + 1 ) e. NN )
3		nnmulcl 8878	. . . . . 6 |- ( ( k e. NN /\ ( k + 1 ) e. NN ) -> ( k x. ( k + 1 ) ) e. NN )
4	2, 3	mpdan 418	. . . . 5 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. NN )
5	4	nncnd 8871	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. CC )
6	4	nnap0d 8903	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) # 0 )
7	1, 5, 6	divrecapd 8689	. . 3 |- ( k e. NN -> ( 2 / ( k x. ( k + 1 ) ) ) = ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
8	7	sumeq2i 11305	. 2 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) )
9		nnuz 9501	. . . . 5 |- NN = ( ZZ>= ` 1 )
10		1zzd 9218	. . . . 5 |- ( T. -> 1 e. ZZ )
11		simpr 109	. . . . . 6 |- ( ( T. /\ k e. NN ) -> k e. NN )
12	4	adantl 275	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( k x. ( k + 1 ) ) e. NN )
13	12	nnrecred 8904	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
14		id 19	. . . . . . . . 9 |- ( n = k -> n = k )
15		oveq1 5849	. . . . . . . . 9 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
16	14, 15	oveq12d 5860	. . . . . . . 8 |- ( n = k -> ( n x. ( n + 1 ) ) = ( k x. ( k + 1 ) ) )
17	16	oveq2d 5858	. . . . . . 7 |- ( n = k -> ( 1 / ( n x. ( n + 1 ) ) ) = ( 1 / ( k x. ( k + 1 ) ) ) )
18		eqid 2165	. . . . . . 7 |- ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) = ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) )
19	17, 18	fvmptg 5562	. . . . . 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 ) ) ) )
20	11, 13, 19	syl2anc 409	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ` k ) = ( 1 / ( k x. ( k + 1 ) ) ) )
21	4	nnrecred 8904	. . . . . . 7 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
22	21	recnd 7927	. . . . . 6 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. CC )
23	22	adantl 275	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. CC )
24	18	trireciplem 11441	. . . . . . 7 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1
25	24	a1i 9	. . . . . 6 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1 )
26		climrel 11221	. . . . . . 7 |- Rel ~~>
27	26	releldmi 4843	. . . . . 6 |- ( seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1 -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) e. dom ~~> )
28	25, 27	syl 14	. . . . 5 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) e. dom ~~> )
29		2cnd 8930	. . . . 5 |- ( T. -> 2 e. CC )
30	9, 10, 20, 23, 28, 29	isummulc2 11367	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
31	9, 10, 20, 23, 25	isumclim 11362	. . . . 5 |- ( T. -> sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) = 1 )
32	31	oveq2d 5858	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
33	30, 32	eqtr3d 2200	. . 3 |- ( T. -> sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
34	33	mptru 1352	. 2 |- sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 )
35		2t1e2 9010	. 2 |- ( 2 x. 1 ) = 2
36	8, 34, 35	3eqtri 2190	1 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2

Syntax hints:   /\ wa 103   = wceq 1343   T. wtru 1344   e. wcel 2136   class class class wbr 3982   |-> cmpt 4043   dom cdm 4604   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   1c1 7754   + caddc 7756   x. cmul 7758   / cdiv 8568   NNcn 8857   2c2 8908   seqcseq 10380   ~~> cli 11219   sum_csu 11294

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by: (None)