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Mirrors > Home > ILE Home > Th. List > trirecip | 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 8921 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) | |
2 | peano2nn 8860 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
3 | nnmulcl 8869 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) | |
4 | 2, 3 | mpdan 418 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
5 | 4 | nncnd 8862 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℂ) |
6 | 4 | nnap0d 8894 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) # 0) |
7 | 1, 5, 6 | divrecapd 8680 | . . 3 ⊢ (𝑘 ∈ ℕ → (2 / (𝑘 · (𝑘 + 1))) = (2 · (1 / (𝑘 · (𝑘 + 1))))) |
8 | 7 | sumeq2i 11291 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) |
9 | nnuz 9492 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 9209 | . . . . 5 ⊢ (⊤ → 1 ∈ ℤ) | |
11 | simpr 109 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
12 | 4 | adantl 275 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
13 | 12 | nnrecred 8895 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
14 | id 19 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | |
15 | oveq1 5843 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) | |
16 | 14, 15 | oveq12d 5854 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑛 + 1)) = (𝑘 · (𝑘 + 1))) |
17 | 16 | oveq2d 5852 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑘 · (𝑘 + 1)))) |
18 | eqid 2164 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | |
19 | 17, 18 | fvmptg 5556 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
20 | 11, 13, 19 | syl2anc 409 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
21 | 4 | nnrecred 8895 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
22 | 21 | recnd 7918 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
23 | 22 | adantl 275 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
24 | 18 | trireciplem 11427 | . . . . . . 7 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 |
25 | 24 | a1i 9 | . . . . . 6 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1) |
26 | climrel 11207 | . . . . . . 7 ⊢ Rel ⇝ | |
27 | 26 | releldmi 4837 | . . . . . 6 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ∈ dom ⇝ ) |
28 | 25, 27 | syl 14 | . . . . 5 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ∈ dom ⇝ ) |
29 | 2cnd 8921 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
30 | 9, 10, 20, 23, 28, 29 | isummulc2 11353 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1))))) |
31 | 9, 10, 20, 23, 25 | isumclim 11348 | . . . . 5 ⊢ (⊤ → Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1))) = 1) |
32 | 31 | oveq2d 5852 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
33 | 30, 32 | eqtr3d 2199 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
34 | 33 | mptru 1351 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1) |
35 | 2t1e2 9001 | . 2 ⊢ (2 · 1) = 2 | |
36 | 8, 34, 35 | 3eqtri 2189 | 1 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 class class class wbr 3976 ↦ cmpt 4037 dom cdm 4598 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 1c1 7745 + caddc 7747 · cmul 7749 / cdiv 8559 ℕcn 8848 2c2 8899 seqcseq 10370 ⇝ cli 11205 Σcsu 11280 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: (None) |
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