Steven R. Dunbar
Department of Mathematics
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University of Nebraska-Lincoln
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Topics in
Probability Theory and Stochastic Processes
Steven R. Dunbar

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Stirling’s Formula from the Sum of Average Differences

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Rating

Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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Section Starter Question

Section Starter Question

How would you estimate ln k k12k+12 ln tdt graphically? How would you use Taylor Series to estimate the difference? Compare that estimate with the estimates from the Trapezoidal Approximation of ln k k12k+12 ln tdt, using the ideas from Stirling’s Formula from Wallis’ Formula and the Trapezoidal Approximation..

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Key Concepts

Key Concepts

  1. Stirling’s Formula as an asymptotic limit follows from estimation of the difference of ln k and the average of ln t on [k 12,k + 12]. Taylor’s Theorem estimates the difference of the mid-value and the average to be O(1k2).

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Vocabulary

Vocabulary

  1. The average of a function f over the interval [a,b] is
    1 b aabf(t) dt.

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Mathematical Ideas

Mathematical Ideas

Stirling’s Approximation

Theorem 1 (Stirling’s Approximation). For each n > 0

n! = 2πnn+12en(1 + ϵ n).

where there exists a real constant A so that |ϵn| < A n .

This is equivalent to n! 2πnn+12en.

Proof. First the proof establishes that there exists a c1 so that log(n!) = c1 + (n + 12) log n n + O(1n).

Expanding the logarithm of n!

log(n!) = k=1n log k = k=1n log k k12k+12 log tdt +k12k+12 log tdt =12n+12 log tdt + k=1n log k k12k+12 log tdt.

Integrating by parts

12n+12 log tdt = t log t t 12n+12 = n + 1 2 log n + 1 2 n + 1 2 1 2 log 1 2 1 2 .

Rewrite the expression log(n + 12) as

log n + 1 2 = log n + log 1 + 1 2n = log n + 1 2n + O 1 n2 .

Combining

12n+12 log tdt = n + 1 2 log n + n + 1 2 1 2n n + 1 2 + 1 2 log 1 2 1 2 + O 1 n = c1 + n + 1 2 log n n + O 1 n

where the constant c1 = 1 2 log 1 2 1 2.

Now consider individual terms in the summation:

log k k12k+12 log tdt = log k t log t t k12k+12 = log k k + 1 2 log k + 1 2 + k 1 2 log k 1 2 + 1 = log k 1 2 log k + 1 2 + log k 1 2 + k log k 1 2 log k + 1 2 + 1 = 1 2 log 1 k2 + log k2 1 4 + k log k(1 1 2k) k(1 + 1 2k) + 1 = 1 2 log(1 4k2) k log 1 + 1 2k log 1 1 2k + 1 = O 1 k2 k 1 2k 1 8k2 + O 1 k3 1 2k 1 8k2 + O 1 k3 + 1 = O 1 k2 k 1 k 1 8k2 + 1 8k2 + O 1 k3 + 1 = O 1 k2 .

So there exists a value c2 so that

log k k12k+12 log tdt c2 k2.

Then set

c3 = k=1log k k12k+12 log tdt.

The tail of the infinite sum is

n+1log k k12k+12 log tdt n+1c2 k2 n+1 c2 k(k 1) = c2 n .

The last equation shows that 1n(log k log tdt) = c 3 + O 1 n.

Combining all of these equations gives us

log(n!) = n + 1 2 log n n + c1 + c3 + O 1 n = n + 1 2 log n n + c4 + O 1 n.

Note that this is essentially the same form as the conclusion of Lemma 3 in Stirling’s Formula by Euler MacLaurin Summation.. Exponentiating, we have

n! = (nn+12eneC6 )(1 + ϵn)

using eO 1 n = (1 + ϵn).

Recall

Lemma 2 (Wallis’ Formula).

lim n 24n (n!)4 ((2n)!)2(2n + 1) = π 2

Proof. See the proofs in Wallis Formula.. □

As before in the proof of Theorem 5 in Stirling’s Formula by Euler MacLaurin Summation. we can conclude that eC6 = 2π. □

Discussion

The Euler-Maclaurin Sum Formula proof of Stirling’s Formula starts with log(n!) = j=1n log(j). The classic proof expresses this as 0n1 log(1 + x) dx with an error term with the Euler-Maclaurin summation formula. The Euler-Maclaurin summation formula is an extension of the Trapezoidal Approximation. (Alternatively, the Euler-Maclaurin summation formula is a result of the Fundamental Theorem of Calculus, summation by parts, and integration by parts.) This allows us to write

log(n!) = n log(n) n + 1 + 1 2 log(n) +1B1(x) x dxϵn (1)

where

ϵn =nB1(x) x dx

and ϵn 0 as n . Then start from Wallis’ Formula and take logarithms, replacing the logarithms of the factorials with equation (1). This provides an equation for the integral 1B1(x) x dx which is solved for the value log(2π) 1. Then the equation above can be exponentiated to express Stirling’s Formula.

Here the proof also starts with log(n!) = j=1n log(j). Then the proof expresses this as

12n+12 log tdt + k=1n log k k12k+12 log tdt.

Here the error term appears as a sum of terms

log k k12k+12 log tdt

which is the difference of log k and the average of log t on [k 12,k + 12]. Since log t is increasing, it is reasonable to expect the difference should be small. The proof that this difference is O(1k2) is established in this section with a Taylor series expansion. Then

log(n!) = n + 1 2 log n n + C1 + C3 + O 1 n = n + 1 2 log n n + C4 + O 1 n.

Together with Wallis’ Formula, this is now enough to establish Stirling’s Formula.

Sources

This section is adapted from: Lesigne, pages 36-39, [1].

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Problems to Work

Problems to Work for Understanding

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Books

Reading Suggestion:

References

[1]   Emmanuel Lesigne. Heads or Tails: An Introduction to Limit Theorems in Probability, volume 28 of Student Mathematical Library. American Mathematical Society, 2005.

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Links

Outside Readings and Links:

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Last modified: Processed from LATEX source on December 16, 2011