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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 11298 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) | |
2 | peano2nn 11237 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
3 | nnmulcl 11248 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) | |
4 | 2, 3 | mpdan 667 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
5 | 4 | nncnd 11241 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℂ) |
6 | 4 | nnne0d 11270 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ≠ 0) |
7 | 1, 5, 6 | divrecd 11009 | . . 3 ⊢ (𝑘 ∈ ℕ → (2 / (𝑘 · (𝑘 + 1))) = (2 · (1 / (𝑘 · (𝑘 + 1))))) |
8 | 7 | sumeq2i 14636 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) |
9 | nnuz 11929 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 11614 | . . . . 5 ⊢ (⊤ → 1 ∈ ℤ) | |
11 | id 22 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | |
12 | oveq1 6802 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) | |
13 | 11, 12 | oveq12d 6813 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑛 + 1)) = (𝑘 · (𝑘 + 1))) |
14 | 13 | oveq2d 6811 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑘 · (𝑘 + 1)))) |
15 | eqid 2771 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | |
16 | ovex 6826 | . . . . . . 7 ⊢ (1 / (𝑘 · (𝑘 + 1))) ∈ V | |
17 | 14, 15, 16 | fvmpt 6426 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
18 | 17 | adantl 467 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
19 | 4 | nnrecred 11271 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
20 | 19 | recnd 10273 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
21 | 20 | adantl 467 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
22 | 15 | trireciplem 14800 | . . . . . . 7 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1) |
24 | climrel 14430 | . . . . . . 7 ⊢ Rel ⇝ | |
25 | 24 | releldmi 5499 | . . . . . 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 11298 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
28 | 9, 10, 18, 21, 26, 27 | isummulc2 14700 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1))))) |
29 | 9, 10, 18, 21, 23 | isumclim 14695 | . . . . 5 ⊢ (⊤ → Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1))) = 1) |
30 | 29 | oveq2d 6811 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
31 | 28, 30 | eqtr3d 2807 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
32 | 31 | trud 1641 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1) |
33 | 2t1e2 11382 | . 2 ⊢ (2 · 1) = 2 | |
34 | 8, 32, 33 | 3eqtri 2797 | 1 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ⊤wtru 1632 ∈ wcel 2145 class class class wbr 4787 ↦ cmpt 4864 dom cdm 5250 ‘cfv 6030 (class class class)co 6795 ℂcc 10139 1c1 10142 + caddc 10144 · cmul 10146 / cdiv 10889 ℕcn 11225 2c2 11275 seqcseq 13007 ⇝ cli 14422 Σcsu 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-om 7216 df-1st 7318 df-2nd 7319 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-oadd 7720 df-er 7899 df-pm 8015 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-n0 11499 df-z 11584 df-uz 11893 df-rp 12035 df-fz 12533 df-fzo 12673 df-fl 12800 df-seq 13008 df-exp 13067 df-hash 13321 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-rlim 14427 df-sum 14624 |
This theorem is referenced by: (None) |
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