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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 9049 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) | |
2 | peano2nn 8988 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ) | |
3 | nnmulcl 8997 | . . . . . 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 8990 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) ∈ ℂ) |
6 | 4 | nnap0d 9022 | . . . 4 ⊢ (𝑘 ∈ ℕ → (𝑘 · (𝑘 + 1)) # 0) |
7 | 1, 5, 6 | divrecapd 8806 | . . 3 ⊢ (𝑘 ∈ ℕ → (2 / (𝑘 · (𝑘 + 1))) = (2 · (1 / (𝑘 · (𝑘 + 1))))) |
8 | 7 | sumeq2i 11501 | . 2 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1)))) |
9 | nnuz 9622 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 9338 | . . . . 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 9023 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
14 | id 19 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) | |
15 | oveq1 5921 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) | |
16 | 14, 15 | oveq12d 5932 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑛 + 1)) = (𝑘 · (𝑘 + 1))) |
17 | 16 | oveq2d 5930 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑘 · (𝑘 + 1)))) |
18 | eqid 2193 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) | |
19 | 17, 18 | fvmptg 5629 | . . . . . 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 9023 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℝ) |
22 | 21 | recnd 8042 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
23 | 22 | adantl 277 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 · (𝑘 + 1))) ∈ ℂ) |
24 | 18 | trireciplem 11637 | . . . . . . 7 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1 |
25 | 24 | a1i 9 | . . . . . 6 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))) ⇝ 1) |
26 | climrel 11417 | . . . . . . 7 ⊢ Rel ⇝ | |
27 | 26 | releldmi 4897 | . . . . . 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 9049 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
30 | 9, 10, 20, 23, 28, 29 | isummulc2 11563 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = Σ𝑘 ∈ ℕ (2 · (1 / (𝑘 · (𝑘 + 1))))) |
31 | 9, 10, 20, 23, 25 | isumclim 11558 | . . . . 5 ⊢ (⊤ → Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1))) = 1) |
32 | 31 | oveq2d 5930 | . . . 4 ⊢ (⊤ → (2 · Σ𝑘 ∈ ℕ (1 / (𝑘 · (𝑘 + 1)))) = (2 · 1)) |
33 | 30, 32 | eqtr3d 2228 | . . 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 9129 | . 2 ⊢ (2 · 1) = 2 | |
36 | 8, 34, 35 | 3eqtri 2218 | 1 ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ⊤wtru 1365 ∈ wcel 2164 class class class wbr 4029 ↦ cmpt 4090 dom cdm 4657 ‘cfv 5250 (class class class)co 5914 ℂcc 7864 ℝcr 7865 1c1 7867 + caddc 7869 · cmul 7871 / cdiv 8685 ℕcn 8976 2c2 9027 seqcseq 10512 ⇝ cli 11415 Σcsu 11490 |
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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4567 ax-iinf 4618 ax-cnex 7957 ax-resscn 7958 ax-1cn 7959 ax-1re 7960 ax-icn 7961 ax-addcl 7962 ax-addrcl 7963 ax-mulcl 7964 ax-mulrcl 7965 ax-addcom 7966 ax-mulcom 7967 ax-addass 7968 ax-mulass 7969 ax-distr 7970 ax-i2m1 7971 ax-0lt1 7972 ax-1rid 7973 ax-0id 7974 ax-rnegex 7975 ax-precex 7976 ax-cnre 7977 ax-pre-ltirr 7978 ax-pre-ltwlin 7979 ax-pre-lttrn 7980 ax-pre-apti 7981 ax-pre-ltadd 7982 ax-pre-mulgt0 7983 ax-pre-mulext 7984 ax-arch 7985 ax-caucvg 7986 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4621 df-xp 4663 df-rel 4664 df-cnv 4665 df-co 4666 df-dm 4667 df-rn 4668 df-res 4669 df-ima 4670 df-iota 5211 df-fun 5252 df-fn 5253 df-f 5254 df-f1 5255 df-fo 5256 df-f1o 5257 df-fv 5258 df-isom 5259 df-riota 5869 df-ov 5917 df-oprab 5918 df-mpo 5919 df-1st 6188 df-2nd 6189 df-recs 6353 df-irdg 6418 df-frec 6439 df-1o 6464 df-oadd 6468 df-er 6582 df-en 6790 df-dom 6791 df-fin 6792 df-pnf 8050 df-mnf 8051 df-xr 8052 df-ltxr 8053 df-le 8054 df-sub 8186 df-neg 8187 df-reap 8588 df-ap 8595 df-div 8686 df-inn 8977 df-2 9035 df-3 9036 df-4 9037 df-n0 9235 df-z 9312 df-uz 9587 df-q 9679 df-rp 9714 df-fz 10069 df-fzo 10203 df-seqfrec 10513 df-exp 10604 df-ihash 10841 df-shft 10953 df-cj 10980 df-re 10981 df-im 10982 df-rsqrt 11136 df-abs 11137 df-clim 11416 df-sumdc 11491 |
This theorem is referenced by: (None) |
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