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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 2727 | . . 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 45381 | . 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 45378 | . 2 โข (๐ โ โ โ ((๐ตโ๐) โ (๐ตโ(๐ + 1))) = (๐ฝโ๐)) |
8 | 4, 7 | breqtrrd 5170 | 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 1534 โ wcel 2099 class class class wbr 5142 โฆ cmpt 5225 โcfv 6542 (class class class)co 7414 1c1 11125 + caddc 11127 ยท cmul 11129 โ cmin 11460 / cdiv 11887 โcn 12228 2c2 12283 โ0cn0 12488 seqcseq 13984 โcexp 14044 !cfa 14250 โcsqrt 15198 โ cli 15446 eceu 16024 logclog 26462 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-xnn0 12561 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15651 df-ef 16029 df-e 16030 df-sin 16031 df-cos 16032 df-tan 16033 df-pi 16034 df-dvds 16217 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-cmp 23265 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-limc 25769 df-dv 25770 df-ulm 26287 df-log 26464 df-cxp 26465 |
This theorem is referenced by: stirlinglem10 45384 stirlinglem11 45385 |
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