Section 5.10

Stirling's Formula

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Problem 3111

Using Stirling's formula, approximately compute: lg⁡100!\lg 100!.

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Problem 3112

Using Stirling's formula, approximately compute: 1⋅3⋅5⋯19991 \cdot 3 \cdot 5\cdots 1999.

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Problem 3113

Using Stirling's formula, approximately compute: 1⋅3⋅5⋯992⋅4⋅6⋯100\dfrac {1 \cdot 3 \cdot 5\cdots 99}{2 \cdot 4 \cdot 6\cdots 100}.

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Problem 3114

Using Stirling's formula, approximately compute: C10040C_{100}^{40}.

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Problem 3115

Using Stirling's formula, approximately compute: 100!20!⋅30!⋅50!\dfrac {100!}{20! \cdot 30! \cdot 50!}.

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Problem 3116

Using Stirling's formula, approximately compute: ∫01(1−x2)50 dx\displaystyle \int_{0}^{1} (1-x^{2})^{50}\, dx.

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Problem 3117

Using Stirling's formula, approximately compute: ∫02πsin⁡200x dx\displaystyle \int_{0}^{2\pi } \sin^{200}x\, dx.

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Problem 3118

For the product

(2n−1)!!=1⋅3⋅5⋯(2n−1) (2n-1)!!=1 \cdot 3 \cdot 5\cdots (2n-1)

derive an asymptotic formula.

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Problem 3119

If nn is very large, approximately compute C2nnC_{2n}^{n}.

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Problem 3120

Using Stirling's formula find the following limits:

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(a)

lim⁡n→∞n!n2\displaystyle \lim_{n \rightarrow \infty } \sqrt[n^{2}]{n!};

(b)

lim⁡n→∞nn!n\displaystyle \lim_{n \rightarrow \infty } \frac{n}{\sqrt[n]{n!}};

(c)

lim⁡n→∞n(2n−1)!!n\displaystyle \lim_{n \rightarrow \infty } \frac{n}{\sqrt[n]{(2n-1)!!}};

(d)

lim⁡n→∞ln⁡n!ln⁡nn\displaystyle \lim_{n \rightarrow \infty } \frac{\ln n!}{\ln n^{n}}.