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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 9022 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) | |
2 | peano2nn 8961 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
3 | nnmulcl 8970 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) | |
4 | 2, 3 | mpdan 421 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
5 | 4 | nncnd 8963 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℂ) |
6 | 4 | nnap0d 8995 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) # 0) |
7 | 1, 5, 6 | divrecapd 8780 | . . 3 ⊢ (𝑘 ∈ ℕ → (2 / (𝑘 · (𝑘 + 1))) = (2 · (1 / (𝑘 · (𝑘 + 1))))) |
8 | 7 | sumeq2i 11404 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) |
9 | nnuz 9593 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 9310 | . . . . 5 ⊢ (⊤ → 1 ∈ ℤ) | |
11 | simpr 110 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
12 | 4 | adantl 277 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 · (𝑘 + 1)) ∈ ℕ) |
13 | 12 | nnrecred 8996 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
14 | id 19 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | |
15 | oveq1 5903 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) | |
16 | 14, 15 | oveq12d 5914 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑛 + 1)) = (𝑘 · (𝑘 + 1))) |
17 | 16 | oveq2d 5912 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑘 · (𝑘 + 1)))) |
18 | eqid 2189 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | |
19 | 17, 18 | fvmptg 5613 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
20 | 11, 13, 19 | syl2anc 411 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))‘𝑘) = (1 / (𝑘 · (𝑘 + 1)))) |
21 | 4 | nnrecred 8996 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
22 | 21 | recnd 8016 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
23 | 22 | adantl 277 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
24 | 18 | trireciplem 11540 | . . . . . . 7 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 |
25 | 24 | a1i 9 | . . . . . 6 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1) |
26 | climrel 11320 | . . . . . . 7 ⊢ Rel ⇝ | |
27 | 26 | releldmi 4884 | . . . . . 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 9022 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
30 | 9, 10, 20, 23, 28, 29 | isummulc2 11466 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1))))) |
31 | 9, 10, 20, 23, 25 | isumclim 11461 | . . . . 5 ⊢ (⊤ → Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1))) = 1) |
32 | 31 | oveq2d 5912 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
33 | 30, 32 | eqtr3d 2224 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
34 | 33 | mptru 1373 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1) |
35 | 2t1e2 9102 | . 2 ⊢ (2 · 1) = 2 | |
36 | 8, 34, 35 | 3eqtri 2214 | 1 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ⊤wtru 1365 ∈ wcel 2160 class class class wbr 4018 ↦ cmpt 4079 dom cdm 4644 ‘cfv 5235 (class class class)co 5896 ℂcc 7839 ℝcr 7840 1c1 7842 + caddc 7844 · cmul 7846 / cdiv 8659 ℕcn 8949 2c2 9000 seqcseq 10476 ⇝ cli 11318 Σcsu 11393 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-frec 6416 df-1o 6441 df-oadd 6445 df-er 6559 df-en 6767 df-dom 6768 df-fin 6769 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-fz 10039 df-fzo 10173 df-seqfrec 10477 df-exp 10551 df-ihash 10788 df-shft 10856 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-clim 11319 df-sumdc 11394 |
This theorem is referenced by: (None) |
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