![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > trirecip | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
trirecip | ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cnd 12323 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) | |
2 | peano2nn 12257 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
3 | nnmulcl 12269 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) | |
4 | 2, 3 | mpdan 685 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
5 | 4 | nncnd 12261 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℂ) |
6 | 4 | nnne0d 12295 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ≠ 0) |
7 | 1, 5, 6 | divrecd 12026 | . . 3 ⊢ (𝑘 ∈ ℕ → (2 / (𝑘 · (𝑘 + 1))) = (2 · (1 / (𝑘 · (𝑘 + 1))))) |
8 | 7 | sumeq2i 15681 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) |
9 | nnuz 12898 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 12626 | . . . . 5 ⊢ (⊤ → 1 ∈ ℤ) | |
11 | id 22 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | |
12 | oveq1 7426 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) | |
13 | 11, 12 | oveq12d 7437 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑛 + 1)) = (𝑘 · (𝑘 + 1))) |
14 | 13 | oveq2d 7435 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑘 · (𝑘 + 1)))) |
15 | eqid 2725 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | |
16 | ovex 7452 | . . . . . . 7 ⊢ (1 / (𝑘 · (𝑘 + 1))) ∈ V | |
17 | 14, 15, 16 | fvmpt 7004 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
18 | 17 | adantl 480 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
19 | 4 | nnrecred 12296 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
20 | 19 | recnd 11274 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
21 | 20 | adantl 480 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
22 | 15 | trireciplem 15844 | . . . . . . 7 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1) |
24 | climrel 15472 | . . . . . . 7 ⊢ Rel ⇝ | |
25 | 24 | releldmi 5950 | . . . . . 6 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ∈ dom ⇝ ) |
26 | 23, 25 | syl 17 | . . . . 5 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ∈ dom ⇝ ) |
27 | 2cnd 12323 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
28 | 9, 10, 18, 21, 26, 27 | isummulc2 15744 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1))))) |
29 | 9, 10, 18, 21, 23 | isumclim 15739 | . . . . 5 ⊢ (⊤ → Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1))) = 1) |
30 | 29 | oveq2d 7435 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
31 | 28, 30 | eqtr3d 2767 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
32 | 31 | mptru 1540 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1) |
33 | 2t1e2 12408 | . 2 ⊢ (2 · 1) = 2 | |
34 | 8, 32, 33 | 3eqtri 2757 | 1 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 / cdiv 11903 ℕcn 12245 2c2 12300 seqcseq 14002 ⇝ cli 15464 Σcsu 15668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |