![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > geoisumr | Structured version Visualization version GIF version |
Description: The infinite sum of reciprocals 1 + (1 / 𝐴)↑1 + (1 / 𝐴)↑2... is 𝐴 / (𝐴 − 1). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
geoisumr | ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → Σ𝑘 ∈ ℕ0 ((1 / 𝐴)↑𝑘) = (𝐴 / (𝐴 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 11915 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11581 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 0 ∈ ℤ) | |
3 | oveq2 6821 | . . . 4 ⊢ (𝑛 = 𝑘 → ((1 / 𝐴)↑𝑛) = ((1 / 𝐴)↑𝑘)) | |
4 | eqid 2760 | . . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) | |
5 | ovex 6841 | . . . 4 ⊢ ((1 / 𝐴)↑𝑘) ∈ V | |
6 | 3, 4, 5 | fvmpt 6444 | . . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
7 | 6 | adantl 473 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
8 | 0le1 10743 | . . . . . . 7 ⊢ 0 ≤ 1 | |
9 | 0re 10232 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
10 | 1re 10231 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
11 | 9, 10 | lenlti 10349 | . . . . . . 7 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
12 | 8, 11 | mpbi 220 | . . . . . 6 ⊢ ¬ 1 < 0 |
13 | fveq2 6352 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
14 | abs0 14224 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
15 | 13, 14 | syl6eq 2810 | . . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
16 | 15 | breq2d 4816 | . . . . . 6 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
17 | 12, 16 | mtbiri 316 | . . . . 5 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
18 | 17 | necon2ai 2961 | . . . 4 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
19 | reccl 10884 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
20 | 18, 19 | sylan2 492 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → (1 / 𝐴) ∈ ℂ) |
21 | expcl 13072 | . . 3 ⊢ (((1 / 𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) | |
22 | 20, 21 | sylan 489 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) |
23 | simpl 474 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 𝐴 ∈ ℂ) | |
24 | simpr 479 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 1 < (abs‘𝐴)) | |
25 | 23, 24, 7 | georeclim 14802 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))) ⇝ (𝐴 / (𝐴 − 1))) |
26 | 1, 2, 7, 22, 25 | isumclim 14687 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → Σ𝑘 ∈ ℕ0 ((1 / 𝐴)↑𝑘) = (𝐴 / (𝐴 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 0cc0 10128 1c1 10129 < clt 10266 ≤ cle 10267 − cmin 10458 / cdiv 10876 ℕ0cn0 11484 ↑cexp 13054 abscabs 14173 Σcsu 14615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-rlim 14419 df-sum 14616 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |