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Mirrors > Home > ILE Home > Th. List > geoihalfsum | GIF version |
Description: Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 11416. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 11418 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 8904 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ) |
3 | 2ap0 8926 | . . . . 5 ⊢ 2 # 0 | |
4 | 3 | a1i 9 | . . . 4 ⊢ (𝑘 ∈ ℕ → 2 # 0) |
5 | nnz 9186 | . . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
6 | 2, 4, 5 | exprecapd 10559 | . . 3 ⊢ (𝑘 ∈ ℕ → ((1 / 2)↑𝑘) = (1 / (2↑𝑘))) |
7 | 6 | sumeq2i 11261 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) |
8 | halfcn 9047 | . . . 4 ⊢ (1 / 2) ∈ ℂ | |
9 | halfre 9046 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
10 | halfge0 9049 | . . . . . 6 ⊢ 0 ≤ (1 / 2) | |
11 | absid 10971 | . . . . . 6 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2)) | |
12 | 9, 10, 11 | mp2an 423 | . . . . 5 ⊢ (abs‘(1 / 2)) = (1 / 2) |
13 | halflt1 9050 | . . . . 5 ⊢ (1 / 2) < 1 | |
14 | 12, 13 | eqbrtri 3985 | . . . 4 ⊢ (abs‘(1 / 2)) < 1 |
15 | geoisum1 11416 | . . . 4 ⊢ (((1 / 2) ∈ ℂ ∧ (abs‘(1 / 2)) < 1) → Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2)))) | |
16 | 8, 14, 15 | mp2an 423 | . . 3 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = ((1 / 2) / (1 − (1 / 2))) |
17 | 1mhlfehlf 9051 | . . . 4 ⊢ (1 − (1 / 2)) = (1 / 2) | |
18 | 17 | oveq2i 5835 | . . 3 ⊢ ((1 / 2) / (1 − (1 / 2))) = ((1 / 2) / (1 / 2)) |
19 | ax-1cn 7825 | . . . . 5 ⊢ 1 ∈ ℂ | |
20 | 1ap0 8465 | . . . . 5 ⊢ 1 # 0 | |
21 | 19, 1, 20, 3 | divap0i 8633 | . . . 4 ⊢ (1 / 2) # 0 |
22 | 8, 21 | dividapi 8618 | . . 3 ⊢ ((1 / 2) / (1 / 2)) = 1 |
23 | 16, 18, 22 | 3eqtri 2182 | . 2 ⊢ Σ𝑘 ∈ ℕ ((1 / 2)↑𝑘) = 1 |
24 | 7, 23 | eqtr3i 2180 | 1 ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 class class class wbr 3965 ‘cfv 5170 (class class class)co 5824 ℂcc 7730 ℝcr 7731 0cc0 7732 1c1 7733 < clt 7912 ≤ cle 7913 − cmin 8046 # cap 8456 / cdiv 8545 ℕcn 8833 2c2 8884 ↑cexp 10418 abscabs 10897 Σcsu 11250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-ihash 10650 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 |
This theorem is referenced by: trilpolemgt1 13621 trilpolemeq1 13622 redcwlpolemeq1 13636 |
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