Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula from the Sum of Average Differences

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1 Steve R Dubar Departmet of Mathematics 03 Avery Hall Uiversity of Nebraska-Licol Licol, NE Voice: Fax: Topics i Probability Theory ad Stochastic Processes Steve R Dubar Stirlig s Formula from the Sum of Average Differeces Ratig Studet: cotais scees of mild algebra or calculus that may require guidace

2 Sectio Starter Questio How would you estimate l k k+/ l t dt graphically? How would you k / use Taylor Series to estimate the differece? Compare that estimate with the estimates from the Trapezoidal Approximatio of l k k+/ l t dt, usig k / the ideas from Stirlig s Formula from Wallis Formula ad the Trapezoidal Approximatio Key Cocepts Stirlig s Formula as a asymptotic limit follows from estimatio of the differece of l k ad the average of l t o [k /, k + /] Taylor s Theorem estimates the differece of the mid-value ad the average to be O/k Vocabulary The average of a fuctio f over the iterval [a, b] is b a b a ft dt

3 Mathematical Ideas Stirlig s Approximatio Theorem Stirlig s Approximatio For each > 0! = π +/ e + ɛ where there exists a real costat A so that ɛ < A This is equivalet to! π +/ e Proof First the proof establishes that there exists a c R so that log! = c + + / log + O/ Expadig the logarithm of! log! = log k = = Itegratig by parts +/ / k= k= +/ / k+/ k / + = [t log t t] +/ / = [ + log + k= Rewrite the expressio log + / as log + = log + log + Combiig +/ / = k+/ k / k+/ k / + + ] + log + + = c + + log + O 3 [ log = log + + O + + [ log ] ] + O

4 where the costat c = [ log ] Now cosider idividual terms i the summatio: k+/ k / [ ] = t log t t k+/ k / = k + log k + + k = [ log k + + ] + [ k log k + ] + = [ log + log k ] + k [ k 4 k + = [ log 4k k log + log k [ = O k k k 8k + O k 3 k ] 8k + O + k [ 3 = O k k k 8k + ] 8k + O + k 3 = O k So there exists a value c so that k+/ k / c k + k k k ] + ] + The set c 3 = k+/ k= k / 4

5 The tail of the ifiite sum is + k+/ k / + + c k c kk = c The last equatio shows that = c 3 + O Combiig all of these equatios gives us log! = + log + c + c 3 + O = + log + c 4 + O Note that this is essetially the same form as the coclusio of Lemma 3 i Stirlig s Formula by Euler MacLauri Summatio Expoetiatig, we have! = +/ e e C 6 + ɛ usig e O = + ɛ Recall Lemma Wallis Formula lim 4! 4! + = π Proof See the proofs i Wallis Formula As before i the proof of Theorem 5 i Stirlig s Formula by Euler MacLauri Summatio we ca coclude that e C 6 = π Discussio The Euler-Maclauri Sum Formula proof of Stirlig s Formula starts with log! = j= logj The classic proof expresses this as log + x dx 0 with a error term with the Euler-Maclauri summatio formula The Euler- Maclauri summatio formula is a extesio of the Trapezoidal Approximatio Alteratively, the Euler-Maclauri summatio formula is a result of 5

6 the Fudametal Theorem of Calculus, summatio by parts, ad itegratio by parts This allows us to write where log! = log + + log + ɛ = B x x dx B x x dx ɛ ad ɛ 0 as The start from Wallis Formula ad take logarithms, replacig the logarithms of the factorials with equatio This provides a equatio for the itegral B x dx which is solved for the x value log π The the equatio above ca be expoetiated to express Stirlig s Formula Here the proof also starts with log! = j= logj The the proof expresses this as +/ k+/ + / k= Here the error term appears as a sum of terms k+/ k / k / which is the differece of log k ad the average of log t o [k /, k + /] Sice log t is icreasig, it is reasoable to expect the differece should be small The proof that this differece is O/k is established i this sectio with a Taylor series expasio The log! = + log + C + C 3 + O = + log + C 4 + O Together with Wallis Formula, this is ow eough to establish Stirlig s Formula 6

7 Sources This sectio is adapted from: Lesige, pages 36-39, [] Problems to Work for Uderstadig 3 4 Readig Suggestio: Refereces [] Emmauel Lesige Heads or Tails: A Itroductio to Limit Theorems i Probability, volume 8 of Studet Mathematical Library America Mathematical Society, 005 7

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