stirlinglem9 - Mathbox for Glauco Siliprandi

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Theorem stirlinglem9 45532

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.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
stirlinglem9	⊢ (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))))

Distinct variable groups: 𝑘,𝑛 𝑛,𝐾 𝑘,𝑁,𝑛 Allowed substitution hints: 𝐴(𝑘,𝑛) 𝐵(𝑘,𝑛) 𝐽(𝑘,𝑛) 𝐾(𝑘)

**Proof of Theorem stirlinglem9**
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 2725	. . 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 45530	. 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 45527	. 2 ⊢ (𝑁 ∈ ℕ → ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))) = (𝐽‘𝑁))
8	4, 7	breqtrrd 5171	1 ⊢ (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6542 (class class class)co 7415 1c1 11137 + caddc 11139 · cmul 11141 − cmin 11472 / cdiv 11899 ℕcn 12240 2c2 12295 ℕ₀cn0 12500 seqcseq 13996 ↑cexp 14056 !cfa 14262 √csqrt 15210 ⇝ cli 15458 eceu 16036 logclog 26504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-e 16042 df-sin 16043 df-cos 16044 df-tan 16045 df-pi 16046 df-dvds 16229 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-ulm 26329 df-log 26506 df-cxp 26507

This theorem is referenced by: stirlinglem10 45533 stirlinglem11 45534

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