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Mirrors > Home > MPE Home > Th. List > Mathboxes > stirlinglem9 | Structured version Visualization version GIF version |
Description: ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))) is expressed as a limit of a series. This result will be used both to prove that 𝐵 is decreasing and to prove that 𝐵 is bounded (below). It will follow that 𝐵 converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
stirlinglem9.1 | ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) |
stirlinglem9.2 | ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) |
stirlinglem9.3 | ⊢ 𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1)) |
stirlinglem9.4 | ⊢ 𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘)))) |
Ref | Expression |
---|---|
stirlinglem9 | ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stirlinglem9.3 | . . 3 ⊢ 𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1)) | |
2 | stirlinglem9.4 | . . 3 ⊢ 𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘)))) | |
3 | eqid 2820 | . . 3 ⊢ (𝑘 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑((2 · 𝑘) + 1))))) = (𝑘 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑((2 · 𝑘) + 1))))) | |
4 | 1, 2, 3 | stirlinglem7 42450 | . 2 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ (𝐽‘𝑁)) |
5 | stirlinglem9.1 | . . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) | |
6 | stirlinglem9.2 | . . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) | |
7 | 5, 6, 1 | stirlinglem4 42447 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))) = (𝐽‘𝑁)) |
8 | 4, 7 | breqtrrd 5075 | 1 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5047 ↦ cmpt 5127 ‘cfv 6336 (class class class)co 7137 1c1 10519 + caddc 10521 · cmul 10523 − cmin 10851 / cdiv 11278 ℕcn 11619 2c2 11674 ℕ0cn0 11879 seqcseq 13354 ↑cexp 13414 !cfa 13618 √csqrt 14572 ⇝ cli 14821 eceu 15396 logclog 25119 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-inf2 9085 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 ax-addf 10597 ax-mulf 10598 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-iin 4903 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-se 5496 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-of 7390 df-om 7562 df-1st 7670 df-2nd 7671 df-supp 7812 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-2o 8084 df-oadd 8087 df-er 8270 df-map 8389 df-pm 8390 df-ixp 8443 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-fsupp 8815 df-fi 8856 df-sup 8887 df-inf 8888 df-oi 8955 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-9 11689 df-n0 11880 df-xnn0 11950 df-z 11964 df-dec 12081 df-uz 12226 df-q 12331 df-rp 12372 df-xneg 12489 df-xadd 12490 df-xmul 12491 df-ioo 12724 df-ioc 12725 df-ico 12726 df-icc 12727 df-fz 12878 df-fzo 13019 df-fl 13147 df-mod 13223 df-seq 13355 df-exp 13415 df-fac 13619 df-bc 13648 df-hash 13676 df-shft 14406 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-limsup 14808 df-clim 14825 df-rlim 14826 df-sum 15023 df-ef 15401 df-e 15402 df-sin 15403 df-cos 15404 df-tan 15405 df-pi 15406 df-dvds 15588 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-mulr 16557 df-starv 16558 df-sca 16559 df-vsca 16560 df-ip 16561 df-tset 16562 df-ple 16563 df-ds 16565 df-unif 16566 df-hom 16567 df-cco 16568 df-rest 16674 df-topn 16675 df-0g 16693 df-gsum 16694 df-topgen 16695 df-pt 16696 df-prds 16699 df-xrs 16753 df-qtop 16758 df-imas 16759 df-xps 16761 df-mre 16835 df-mrc 16836 df-acs 16838 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-submnd 17935 df-mulg 18203 df-cntz 18425 df-cmn 18886 df-psmet 20515 df-xmet 20516 df-met 20517 df-bl 20518 df-mopn 20519 df-fbas 20520 df-fg 20521 df-cnfld 20524 df-top 21480 df-topon 21497 df-topsp 21519 df-bases 21532 df-cld 21605 df-ntr 21606 df-cls 21607 df-nei 21684 df-lp 21722 df-perf 21723 df-cn 21813 df-cnp 21814 df-haus 21901 df-cmp 21973 df-tx 22148 df-hmeo 22341 df-fil 22432 df-fm 22524 df-flim 22525 df-flf 22526 df-xms 22908 df-ms 22909 df-tms 22910 df-cncf 23464 df-limc 24444 df-dv 24445 df-ulm 24946 df-log 25121 df-cxp 25122 |
This theorem is referenced by: stirlinglem10 42453 stirlinglem11 42454 |
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