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Mirrors > Home > MPE Home > Th. List > geoihalfsum | Structured version Visualization version GIF version |
Description: Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 15223. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 15225 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by AV, 9-Jul-2022.) |
Ref | Expression |
---|---|
geoihalfsum | ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11700 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) |
3 | 2ne0 11729 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ≠ 0) |
5 | nnz 11992 | . . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
6 | 2, 4, 5 | exprecd 13506 | . . 3 ⊢ (𝑘 ∈ ℕ → ((1 / 2)↑𝑘) = (1 / (2↑𝑘))) |
7 | 6 | sumeq2i 15044 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) |
8 | halfcn 11840 | . . . 4 ⊢ (1 / 2) ∈ ℂ | |
9 | halfre 11839 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
10 | halfge0 11842 | . . . . . 6 ⊢ 0 ≤ (1 / 2) | |
11 | absid 14644 | . . . . . 6 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2)) | |
12 | 9, 10, 11 | mp2an 688 | . . . . 5 ⊢ (abs‘(1 / 2)) = (1 / 2) |
13 | halflt1 11843 | . . . . 5 ⊢ (1 / 2) < 1 | |
14 | 12, 13 | eqbrtri 5078 | . . . 4 ⊢ (abs‘(1 / 2)) < 1 |
15 | geoisum1 15223 | . . . 4 ⊢ (((1 / 2) ∈ ℂ ∧ (abs‘(1 / 2)) < 1) → Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2)))) | |
16 | 8, 14, 15 | mp2an 688 | . . 3 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2))) |
17 | 1mhlfehlf 11844 | . . . 4 ⊢ (1 − (1 / 2)) = (1 / 2) | |
18 | 17 | oveq2i 7156 | . . 3 ⊢ ((1 / 2) / (1 − (1 / 2))) = ((1 / 2) / (1 / 2)) |
19 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | ax-1ne0 10594 | . . . . 5 ⊢ 1 ≠ 0 | |
21 | 19, 1, 20, 3 | divne0i 11376 | . . . 4 ⊢ (1 / 2) ≠ 0 |
22 | 8, 21 | dividi 11361 | . . 3 ⊢ ((1 / 2) / (1 / 2)) = 1 |
23 | 16, 18, 22 | 3eqtri 2845 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = 1 |
24 | 7, 23 | eqtr3i 2843 | 1 ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 < clt 10663 ≤ cle 10664 − cmin 10858 / cdiv 11285 ℕcn 11626 2c2 11680 ↑cexp 13417 abscabs 14581 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 |
This theorem is referenced by: omssubadd 31457 |
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