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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 12904 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12610 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 0 ∈ ℤ) | |
3 | oveq2 7434 | . . . 4 ⊢ (𝑛 = 𝑘 → ((1 / 𝐴)↑𝑛) = ((1 / 𝐴)↑𝑘)) | |
4 | eqid 2728 | . . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) | |
5 | ovex 7459 | . . . 4 ⊢ ((1 / 𝐴)↑𝑘) ∈ V | |
6 | 3, 4, 5 | fvmpt 7010 | . . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
7 | 6 | adantl 480 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
8 | 0le1 11777 | . . . . . . 7 ⊢ 0 ≤ 1 | |
9 | 0re 11256 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
10 | 1re 11254 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
11 | 9, 10 | lenlti 11374 | . . . . . . 7 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
12 | 8, 11 | mpbi 229 | . . . . . 6 ⊢ ¬ 1 < 0 |
13 | fveq2 6902 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
14 | abs0 15274 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
15 | 13, 14 | eqtrdi 2784 | . . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
16 | 15 | breq2d 5164 | . . . . . 6 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
17 | 12, 16 | mtbiri 326 | . . . . 5 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
18 | 17 | necon2ai 2967 | . . . 4 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
19 | reccl 11919 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
20 | 18, 19 | sylan2 591 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → (1 / 𝐴) ∈ ℂ) |
21 | expcl 14086 | . . 3 ⊢ (((1 / 𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) | |
22 | 20, 21 | sylan 578 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) |
23 | simpl 481 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 𝐴 ∈ ℂ) | |
24 | simpr 483 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 1 < (abs‘𝐴)) | |
25 | 23, 24, 7 | georeclim 15860 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))) ⇝ (𝐴 / (𝐴 − 1))) |
26 | 1, 2, 7, 22, 25 | isumclim 15745 | 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ℂcc 11146 0cc0 11148 1c1 11149 < clt 11288 ≤ cle 11289 − cmin 11484 / cdiv 11911 ℕ0cn0 12512 ↑cexp 14068 abscabs 15223 Σcsu 15674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-rlim 15475 df-sum 15675 |
This theorem is referenced by: (None) |
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