![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > stirlinglem9 | Structured version Visualization version GIF version |
Description: ((๐ตโ๐) โ (๐ตโ(๐ + 1))) is expressed as a limit of a series. This result will be used both to prove that ๐ต is decreasing and to prove that ๐ต is bounded (below). It will follow that ๐ต converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
stirlinglem9.1 | โข ๐ด = (๐ โ โ โฆ ((!โ๐) / ((โโ(2 ยท ๐)) ยท ((๐ / e)โ๐)))) |
stirlinglem9.2 | โข ๐ต = (๐ โ โ โฆ (logโ(๐ดโ๐))) |
stirlinglem9.3 | โข ๐ฝ = (๐ โ โ โฆ ((((1 + (2 ยท ๐)) / 2) ยท (logโ((๐ + 1) / ๐))) โ 1)) |
stirlinglem9.4 | โข ๐พ = (๐ โ โ โฆ ((1 / ((2 ยท ๐) + 1)) ยท ((1 / ((2 ยท ๐) + 1))โ(2 ยท ๐)))) |
Ref | Expression |
---|---|
stirlinglem9 | โข (๐ โ โ โ seq1( + , ๐พ) โ ((๐ตโ๐) โ (๐ตโ(๐ + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stirlinglem9.3 | . . 3 โข ๐ฝ = (๐ โ โ โฆ ((((1 + (2 ยท ๐)) / 2) ยท (logโ((๐ + 1) / ๐))) โ 1)) | |
2 | stirlinglem9.4 | . . 3 โข ๐พ = (๐ โ โ โฆ ((1 / ((2 ยท ๐) + 1)) ยท ((1 / ((2 ยท ๐) + 1))โ(2 ยท ๐)))) | |
3 | eqid 2725 | . . 3 โข (๐ โ โ0 โฆ (2 ยท ((1 / ((2 ยท ๐) + 1)) ยท ((1 / ((2 ยท ๐) + 1))โ((2 ยท ๐) + 1))))) = (๐ โ โ0 โฆ (2 ยท ((1 / ((2 ยท ๐) + 1)) ยท ((1 / ((2 ยท ๐) + 1))โ((2 ยท ๐) + 1))))) | |
4 | 1, 2, 3 | stirlinglem7 45530 | . 2 โข (๐ โ โ โ seq1( + , ๐พ) โ (๐ฝโ๐)) |
5 | stirlinglem9.1 | . . 3 โข ๐ด = (๐ โ โ โฆ ((!โ๐) / ((โโ(2 ยท ๐)) ยท ((๐ / e)โ๐)))) | |
6 | stirlinglem9.2 | . . 3 โข ๐ต = (๐ โ โ โฆ (logโ(๐ดโ๐))) | |
7 | 5, 6, 1 | stirlinglem4 45527 | . 2 โข (๐ โ โ โ ((๐ตโ๐) โ (๐ตโ(๐ + 1))) = (๐ฝโ๐)) |
8 | 4, 7 | breqtrrd 5171 | 1 โข (๐ โ โ โ seq1( + , ๐พ) โ ((๐ตโ๐) โ (๐ตโ(๐ + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 class class class wbr 5143 โฆ cmpt 5226 โcfv 6542 (class class class)co 7415 1c1 11137 + caddc 11139 ยท cmul 11141 โ cmin 11472 / cdiv 11899 โcn 12240 2c2 12295 โ0cn0 12500 seqcseq 13996 โcexp 14056 !cfa 14262 โcsqrt 15210 โ 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 |
Copyright terms: Public domain | W3C validator |