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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 15237. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 15239 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 11715 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) |
3 | 2ne0 11744 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ≠ 0) |
5 | nnz 12007 | . . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
6 | 2, 4, 5 | exprecd 13521 | . . 3 ⊢ (𝑘 ∈ ℕ → ((1 / 2)↑𝑘) = (1 / (2↑𝑘))) |
7 | 6 | sumeq2i 15058 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) |
8 | halfcn 11855 | . . . 4 ⊢ (1 / 2) ∈ ℂ | |
9 | halfre 11854 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
10 | halfge0 11857 | . . . . . 6 ⊢ 0 ≤ (1 / 2) | |
11 | absid 14658 | . . . . . 6 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2)) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (abs‘(1 / 2)) = (1 / 2) |
13 | halflt1 11858 | . . . . 5 ⊢ (1 / 2) < 1 | |
14 | 12, 13 | eqbrtri 5089 | . . . 4 ⊢ (abs‘(1 / 2)) < 1 |
15 | geoisum1 15237 | . . . 4 ⊢ (((1 / 2) ∈ ℂ ∧ (abs‘(1 / 2)) < 1) → Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2)))) | |
16 | 8, 14, 15 | mp2an 690 | . . 3 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2))) |
17 | 1mhlfehlf 11859 | . . . 4 ⊢ (1 − (1 / 2)) = (1 / 2) | |
18 | 17 | oveq2i 7169 | . . 3 ⊢ ((1 / 2) / (1 − (1 / 2))) = ((1 / 2) / (1 / 2)) |
19 | ax-1cn 10597 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | ax-1ne0 10608 | . . . . 5 ⊢ 1 ≠ 0 | |
21 | 19, 1, 20, 3 | divne0i 11390 | . . . 4 ⊢ (1 / 2) ≠ 0 |
22 | 8, 21 | dividi 11375 | . . 3 ⊢ ((1 / 2) / (1 / 2)) = 1 |
23 | 16, 18, 22 | 3eqtri 2850 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = 1 |
24 | 7, 23 | eqtr3i 2848 | 1 ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 ↑cexp 13432 abscabs 14595 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 |
This theorem is referenced by: omssubadd 31560 |
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