Re: Sum of Log

[llopes@xxxxxxxxxxxxx]

Sauda,c~oes,

Um amigo me sugeriu a seguinte lista :

Achei a tal lista de matematica:

- Entre no site de newsgroup - http://www.deja.com
- escolha a opção sci (science)
- escolha sci.math

Pelos assuntos de algumas mensagens, como a que 
segue (de novo a fun,c~ao Gama), a lista merece 
uma olhada.

[ ]'s
Lu'is

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Subject: Re: Sum of Log
From: jmccarty@sun1307.ssd.usa.alcatel.com (Mike Mccarty Sr)
Date: 2000/09/22
Newsgroups: sci.math
In article <8qakep$846@mcmail.cis.McMaster.CA>,
Zdislav V. Kovarik <kovarik@mcmail.cis.McMaster.CA> wrote:
)In article <k9qfwomjoqwx@forum.mathforum.com>,
)C.Y. Kwong <cy_aries@yahoo.com> wrote:
):I tried to find the number of digits in the expansion of 2000! by
):finding the sum of log1, log2, log3,..., log2000 (common log) with a
):programmable calculator. The sum found is about 5735.52057.
):(So 2000! is about 3.3*10^5735, with 5736 digits)
):
):Then I tried to get an approximate value of this by finding the
):integrals
):Int(1 to 2000)[ln(x)]dx/ln(10),
):Int(2 to 2001)[ln(x)]dx/ln(10), and
):Int(1.5 to 2000.5)[ln(x)]dx/ln(10) .
):The value of the last one is about 5735.50887.
):I think this is a good approximation of the sum.
):
):Is this a well known result?
):Can anybody support this by mathematics?
):
):Thanks.
)
)On the introductory level, you can compare the integral with the lower and
)upper integral sums using intervals of length 1. A result (you are invited
)to check it) is, after some easy algebra,
)
)    n*ln(n) - n + 1 < ln(n!) < (n+1)*ln(n) - n + 1
)
)so if you fiddle around with the bounds of integration, you can get a
)closer fit. 
)
)On a higher level, a refined estimate was found by Stirling during the
)early times of Calculus. Find out about Stirling's Formula.
)
)Good luck, ZVK(Slavek).

Even better, 

Gamma(x) ~ sqrt(2*PI)*x^(x-1/2)*exp(-x+1/(12*x)-1/(360*x^3)+1/(1260*x^5)-1/(1680*x^7)...)

and n! = Gamma(x+1).

On another level, (5636 digits) 2000! =

              331627509245063324117539338057632403
82811172081057803945719354370603807790560082240027
32308597325922554023529412258341092580848174152937
96131386633526343688905634058556163940605117252571
87064785639354404540524395746703767410872297043468
41583437524315808775336451274879954368592474080324
08946561507233250652797655757179671536718689359056
11281587160171723265715611000421401242043384257371
27001758835477968999212835289966658534055798549036
57366350133386550401172012152635488038268152152246
92099520603156441856548067594649705155228820523489
99957264508140655366789695321014676226713320268315
52205194494461618239275204026529722631502574752048
29606475092739416585628353177957448287631459645037
39913273341772636088524900935066216101444597094127
07821313732563831572302019949914958316470942774473
87032798554967429860883937632682415247883438746959
58292577405745398375015858154681362942179499723998
13599481016556563876034227312912250384709872909626
62246197107660593155020189513558316535787149229091
67790497022470946119376077851651106844322559056487
36266530377384650390788049524600712549402614566072
25413630275491367158340609783107494528221749078134
77096932415561113398280513586006905946199652573107
41177081519922564516778571458056602185654760952377
46301667942248844448579834980154803262082989096585
73817518886193766928282798884535846398965942139529
84465291092009103710046149449915828588050761867924
94638518087987451289140801934007462592005709872957
85996436506558956124102310186905560603087836291105
05601245908998383410799367902052076858669183477906
55854470014869265692463193333761242809742006717284
63619392496986284687199934503938893672704871271727
34561700354867477509102955523953547941107421913301
35681954109194146276641754216158762526285808980122
24438902486771820549594157519917012717675717874958
61619665931878855141835782092601482071777331735396
03430496908207058995870138198081303559016076290838
85745612882176981361824835767392183031184147191339
86892842344000779246691209766731651433494437473235
63657204884447833185494169303012453167623274536787
93228474738244850922831399525097325059791270310476
83601481191102229253372697693823670057565612400290
57604385285290293760647953345817966612383960526254
91071866638693547661084550461981020840506358276765
26589492393249519685954171672419329530683673495544
00458635983816104305944982662753060542358075589410
82788804278259510898806354105679179509740177806887
82869810219010900148352061688883720250310665922068
60148364983053278208826353655804360568678128416921
71330471411763121758957771226375847531235172309905
49829210134687304205898014418063875382664169897704
23775940628087725370226542653058086237930142267582
11871435029186376363403001732518182620760397473695
95202642632364145446851113427202150458383851010136
94131303485622191663162389263276581535501127630782
50599691588245334574354378636831737306732965893551
99694458236873508830278657700879749889992343555566
24068283476378468518384497364887395247510322422211
05612012958296571913681086938254757641188868793467
25191246192151144738836269591643672490071653428228
15266124780046392254494517036372362794075778454209
10483054616561906221742869816029733240465202019928
13854882681951007282869701070737500927666487502174
77537274235150874824672027417003158112280589617812
21607474379475109506209385566745812525183766821577
12807861499255876132352950422346387878954850885764
46613629039412766597804420209228133798711590089626
48789424132104549250035666706329094415793729867434
21470507213588932019580723064781498429522595589012
75482397177332572291032576092979073329954505638836
26404746502450808094691160726320874941439730007041
11418595530278827357654819182002449697761111346318
19528276159096418979095811733862720608891043294524
49785351470141124421430554860896395783783473253235
95763291438925288393986256273242862775563140463830
38916842163311344563630957196597846633855149231619
63356753551384034258041629198378222669095217701531
75338730284610841886554138329171951332117895728541
66208482368281793251293123752154192697026970329947
76438233864830088715303734056663838682940884877307
21762268849023084934661194260180272613802108005078
21574100605484820134785957810277070778065551277254
05016743323960662532164150048087724030476119290322
10154385353138685538486425570790795341176519571188
68373988068389579274374968349814292329219630977709
01439368436553333593078201813129934550242060445633
40578606962471961505603394899523321800434359967256
62392719643540287205547501207985433197067479731312
68135236537440856622632067688375851327828962523332
84341812977624697079543436003492343159239674763638
91211528540665778364621391124744705125522634270123
95270181270454916480459322481088586746009523067931
75967755581011679940005249806303763141344412269037
03498735579991600925924807505248554156826628176081
54463083054066774126301244418642041083731190931300
01154470560277773724378067188899770851056727276781
24719883285769584421758889516046786820481001004781
64623582208385324881342708340798684866321627202088
23308727819085378845469131556021728873121907393965
20926022910147752708093086536497985855401057745027
92898146036884318215086372462169678722821693473705
99286277112447690920902988320166830170273420259765
67170986331121634950217126442682711965026405422823
17596308744753018471940955242634114984695080733900
80000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
-- 
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I can explain it for you, but I can't understand it for you.
I don't speak for Alcatel      <- They make me say that.


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