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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 11598 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11266 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 0 ∈ ℤ) | |
3 | oveq2 6557 | . . . 4 ⊢ (𝑛 = 𝑘 → ((1 / 𝐴)↑𝑛) = ((1 / 𝐴)↑𝑘)) | |
4 | eqid 2610 | . . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛)) | |
5 | ovex 6577 | . . . 4 ⊢ ((1 / 𝐴)↑𝑘) ∈ V | |
6 | 3, 4, 5 | fvmpt 6191 | . . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
7 | 6 | adantl 481 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))‘𝑘) = ((1 / 𝐴)↑𝑘)) |
8 | 0le1 10430 | . . . . . . 7 ⊢ 0 ≤ 1 | |
9 | 0re 9919 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
10 | 1re 9918 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
11 | 9, 10 | lenlti 10036 | . . . . . . 7 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
12 | 8, 11 | mpbi 219 | . . . . . 6 ⊢ ¬ 1 < 0 |
13 | fveq2 6103 | . . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0)) | |
14 | abs0 13873 | . . . . . . . 8 ⊢ (abs‘0) = 0 | |
15 | 13, 14 | syl6eq 2660 | . . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0) |
16 | 15 | breq2d 4595 | . . . . . 6 ⊢ (𝐴 = 0 → (1 < (abs‘𝐴) ↔ 1 < 0)) |
17 | 12, 16 | mtbiri 316 | . . . . 5 ⊢ (𝐴 = 0 → ¬ 1 < (abs‘𝐴)) |
18 | 17 | necon2ai 2811 | . . . 4 ⊢ (1 < (abs‘𝐴) → 𝐴 ≠ 0) |
19 | reccl 10571 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
20 | 18, 19 | sylan2 490 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → (1 / 𝐴) ∈ ℂ) |
21 | expcl 12740 | . . 3 ⊢ (((1 / 𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) | |
22 | 20, 21 | sylan 487 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) ∧ 𝑘 ∈ ℕ0) → ((1 / 𝐴)↑𝑘) ∈ ℂ) |
23 | simpl 472 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 𝐴 ∈ ℂ) | |
24 | simpr 476 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → 1 < (abs‘𝐴)) | |
25 | 23, 24, 7 | georeclim 14442 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑛))) ⇝ (𝐴 / (𝐴 − 1))) |
26 | 1, 2, 7, 22, 25 | isumclim 14330 | 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 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕ0cn0 11169 ↑cexp 12722 abscabs 13822 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 |
This theorem is referenced by: (None) |
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