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Mirrors > Home > MPE Home > Th. List > leibpisum | Structured version Visualization version GIF version |
Description: The Leibniz formula for π. This version of leibpi 26970 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.) |
Ref | Expression |
---|---|
leibpisum | ⊢ Σ𝑛 ∈ ℕ0 ((-1↑𝑛) / ((2 · 𝑛) + 1)) = (π / 4) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12916 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12622 | . . 3 ⊢ (⊤ → 0 ∈ ℤ) | |
3 | oveq2 7432 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (-1↑𝑘) = (-1↑𝑛)) | |
4 | oveq2 7432 | . . . . . . 7 ⊢ (𝑘 = 𝑛 → (2 · 𝑘) = (2 · 𝑛)) | |
5 | 4 | oveq1d 7439 | . . . . . 6 ⊢ (𝑘 = 𝑛 → ((2 · 𝑘) + 1) = ((2 · 𝑛) + 1)) |
6 | 3, 5 | oveq12d 7442 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1))) |
7 | eqid 2726 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1))) = (𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1))) | |
8 | ovex 7457 | . . . . 5 ⊢ ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ V | |
9 | 6, 7, 8 | fvmpt 7009 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1)))‘𝑛) = ((-1↑𝑛) / ((2 · 𝑛) + 1))) |
10 | 9 | adantl 480 | . . 3 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1)))‘𝑛) = ((-1↑𝑛) / ((2 · 𝑛) + 1))) |
11 | neg1rr 12379 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
12 | reexpcl 14098 | . . . . . . 7 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ0) → (-1↑𝑛) ∈ ℝ) | |
13 | 11, 12 | mpan 688 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (-1↑𝑛) ∈ ℝ) |
14 | 2nn0 12541 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
15 | nn0mulcl 12560 | . . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0) | |
16 | 14, 15 | mpan 688 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (2 · 𝑛) ∈ ℕ0) |
17 | nn0p1nn 12563 | . . . . . . 7 ⊢ ((2 · 𝑛) ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ) |
19 | 13, 18 | nndivred 12318 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ) |
20 | 19 | recnd 11292 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ) |
21 | 20 | adantl 480 | . . 3 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ) |
22 | 7 | leibpi 26970 | . . . 4 ⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1)))) ⇝ (π / 4) |
23 | 22 | a1i 11 | . . 3 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ ((-1↑𝑘) / ((2 · 𝑘) + 1)))) ⇝ (π / 4)) |
24 | 1, 2, 10, 21, 23 | isumclim 15761 | . 2 ⊢ (⊤ → Σ𝑛 ∈ ℕ0 ((-1↑𝑛) / ((2 · 𝑛) + 1)) = (π / 4)) |
25 | 24 | mptru 1541 | 1 ⊢ Σ𝑛 ∈ ℕ0 ((-1↑𝑛) / ((2 · 𝑛) + 1)) = (π / 4) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 class class class wbr 5153 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 -cneg 11495 / cdiv 11921 ℕcn 12264 2c2 12319 4c4 12321 ℕ0cn0 12524 seqcseq 14021 ↑cexp 14081 ⇝ cli 15486 Σcsu 15690 πcpi 16068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-tan 16073 df-pi 16074 df-dvds 16257 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-t1 23309 df-haus 23310 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-ulm 26406 df-log 26583 df-atan 26895 |
This theorem is referenced by: (None) |
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