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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 15098. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 15100 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 11518 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) |
3 | 2ne0 11554 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ≠ 0) |
5 | nnz 11820 | . . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
6 | 2, 4, 5 | exprecd 13336 | . . 3 ⊢ (𝑘 ∈ ℕ → ((1 / 2)↑𝑘) = (1 / (2↑𝑘))) |
7 | 6 | sumeq2i 14919 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) |
8 | halfcn 11665 | . . . 4 ⊢ (1 / 2) ∈ ℂ | |
9 | halfre 11664 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
10 | halfge0 11667 | . . . . . 6 ⊢ 0 ≤ (1 / 2) | |
11 | absid 14520 | . . . . . 6 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2)) | |
12 | 9, 10, 11 | mp2an 679 | . . . . 5 ⊢ (abs‘(1 / 2)) = (1 / 2) |
13 | halflt1 11668 | . . . . 5 ⊢ (1 / 2) < 1 | |
14 | 12, 13 | eqbrtri 4951 | . . . 4 ⊢ (abs‘(1 / 2)) < 1 |
15 | geoisum1 15098 | . . . 4 ⊢ (((1 / 2) ∈ ℂ ∧ (abs‘(1 / 2)) < 1) → Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2)))) | |
16 | 8, 14, 15 | mp2an 679 | . . 3 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2))) |
17 | 1mhlfehlf 11669 | . . . 4 ⊢ (1 − (1 / 2)) = (1 / 2) | |
18 | 17 | oveq2i 6989 | . . 3 ⊢ ((1 / 2) / (1 − (1 / 2))) = ((1 / 2) / (1 / 2)) |
19 | ax-1cn 10395 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | ax-1ne0 10406 | . . . . 5 ⊢ 1 ≠ 0 | |
21 | 19, 1, 20, 3 | divne0i 11191 | . . . 4 ⊢ (1 / 2) ≠ 0 |
22 | 8, 21 | dividi 11176 | . . 3 ⊢ ((1 / 2) / (1 / 2)) = 1 |
23 | 16, 18, 22 | 3eqtri 2806 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = 1 |
24 | 7, 23 | eqtr3i 2804 | 1 ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ≠ wne 2967 class class class wbr 4930 ‘cfv 6190 (class class class)co 6978 ℂcc 10335 ℝcr 10336 0cc0 10337 1c1 10338 < clt 10476 ≤ cle 10477 − cmin 10672 / cdiv 11100 ℕcn 11441 2c2 11498 ↑cexp 13247 abscabs 14457 Σcsu 14906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-pm 8211 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-fz 12712 df-fzo 12853 df-fl 12980 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-clim 14709 df-rlim 14710 df-sum 14907 |
This theorem is referenced by: omssubadd 31203 |
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