STIRLING'S 1 FORMULA AND ITS APPLICATION
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1 MAT-KOL (Baja Luka) XXIV ()(08) DOI: 075/МК80057A ISSN (o) ISSN (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract: I this papr ar giv th proof of importat Stirlig's formula ad svral hrs itrstig applicatios Ky words ad phrass: Stirlig's formula Taylor sris limit of squc Wirstrass critrium Wallis formula Th proof of th Stirlig's formula Stirlig's formula has th form:! W bgi with th Taylor sris xpasios: x x x x l x x ; 0 () for x Combiig ths two w obtai th Taylor sris xpasio: agai for x I particular for x 3 5 m l x x x x x 3 5 m x whr N w hav: Jams Stirlig (69-77) Scottish mathmaticia
2 MAT-KOL XXIV (08) Š Arslaagić which ca b writt as: l l Th right-had sid is gratr tha It ca b boudd from abov as follows: k 4 k 3 So usig Taylor sris w hav obtaid th doubl iquality: or l l This trasforms by xpotiatig ad dividig by ito: To brig this closr to Stirlig's formula ot that th trm i th middl is qual to:! x! x 58
3 MAT-KOL XXIV (08) Š Arslaagić whr x! a umbr that w wat to prov is qual to 0 I ordr to prov this w writ th abov doubl iquality as: with x x W dduc that th squc x is icrasig Bcaus x is positiv ad dcrasig whil th squc covrgs to ad bcaus (x ) N covrgs by th Wirstrass critrio both x ad x must covrg to th sam limit L W claim that L Bfor provig this ot that: x L x so by th itrmdiat valu proprty thr xists 0 such that x L L x i Th oly thig lft is th computatio of th limit L For this w mploy th Wallis 3 formula: 46 lim 3 5 W rwrit this limit as:! lim! Substitutig! ad! b th formula foud abov givs: L 4 lim lim L 4 L 4 Karl Thodor Withlm Wirstrass (85-897) grma mathmaticia 3 Joh Wallis (66-703) glish mathmaticia 59
4 MAT-KOL XXIV (08) Š Arslaagić Hc L ad Stirlig's formula () is provd Th applicatio of th Stirlig's formula Exampl Prov that th squc ad fid its limit Solutio It uss Stirlig's formula (): a!! ; N is covrgt! with 0 Takig th -th root ad passig to th limit w obtai: lim! W also dduc that lim lim!! Thrfor!! lim lim!! lim lim lim!!! lim lim! Takig th -th root ad passig to th limit w obtai: ad hc! lim! 60
5 MAT-KOL XXIV (08) Š Arslaagić Thus if w st th lim b w obtai From th quality: a! lim lim 0!! b a!! a! a! b!! al lb!! Th right-had sid is a product of thr squcs that ovrg rspctivly to l l ad Thrfor th squc (a ) N covrgs to th limit qd Exampl Prov that l lim! Solutio Usig th doubl iquality with rrgard to Stirlig's formula provd of th H Robbis i [5] w hav: or!! ; N! l l! 6
6 MAT-KOL XXIV (08) Š Arslaagić Bcaus l lim l l lim l l lim ad aalogously l l lim l l lim Usig ow th kow critrio for th covrgc of a squc ad its limit w gt: l lim! qd Exampl 3 Comput l lim! Solutio W will us th rsult of th Exampl : W hav furthr by (): l lim! () 6
7 MAT-KOL XXIV (08) Š Arslaagić lim! lim! lim l lim lim l l l l l l l l lim bcaus lim i l ad from hr: l l lim! lim l l lim! lim i l l lim! qd Rmark I th mathmatical litrary works th approximatio:! (3) has too th am th Stirlig's formula W hav i [4] pag 6 th xampl 8 of th iquality (with th proof): <! < ( N) (4) Th proof follows usig th mathmatical iductio ad th iquality But from (3) ad (4) w gt: 63
8 MAT-KOL XXIV (08) Š Arslaagić i 4 what is xact bcaus 68 ; ad Rfrcs [] Š Arslaagić Mathmatical raig book 8 [Matmatička čitaka 8] Grafičar promt doo Sarajvo 06 [] Š Arslaagić Mathmatical raig book 9 [Matmatička čitaka 9] Grafičar promt doo Sarajvo 07 [3] C Balcau ad MF Dumitrscu Asupra uor limit d siruri Rvista d Matmatica Mariscu-Ghmci Octavia Editura Hoffma Potcoava (Rumuija) ()(07) [4] I I Ljaško AK Bojarčuk JG Gaj ad GP Golovič Spravočo posobi po matmatičskomu aalizu Tom Th scod rvisd ditio Viša škola Kijv 984 [5] H Robbis A rmark o Stirlig's formula Th Amrica Mathmatical Mothly 6()(955) 6-9 [6] D Vlja Combiativ ad discrt mathmatics [Kombiatora i diskrta matmatika] Algoritam Zagrb 00 64
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