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[parent] weaker version of Stirling's approximation (Result)

One can prove a weaker version of Stirling's approximation without appealing to the gamma function. Consider the graph of $ \ln x$ and note that

$\displaystyle \ln(n-1)!\leq \int_1^n \ln x \,\mathrm{d} x\leq \ln n!$
But $ \int \ln x \,\mathrm{d} x=x\ln x-x$, so
$\displaystyle \ln(n-1)!\leq n\ln n-n+1\leq \ln n!$
and thus
$\displaystyle n\ln n-n+1+\ln n\geq\ln(n-1)!+\ln n=\ln n!\geq n\ln n-n+1$
so
$\displaystyle \ln n-1+\frac{1}{n}+\frac{\ln n}{n}\geq\frac{1}{n}\ln n!\geq\ln n-1+\frac{1}{n}$
As $ n$ gets large, the expressions on either end approach $ \ln n-1$, so we have

$\displaystyle \frac{1}{n}\ln n! \approx \ln n - 1$

Multiplying through by $ n$ and exponentiating, we get

$\displaystyle n!\approx n^ne^{-n}$



"weaker version of Stirling's approximation" is owned by rm50.
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Cross-references: graph, gamma function

This is version 4 of weaker version of Stirling's approximation, born on 2006-11-19, modified 2006-11-21.
Object id is 8573, canonical name is WeakerVersionOfStirlingsApproximation.
Accessed 356 times total.

Classification:
AMS MSC41A60 (Approximations and expansions :: Asymptotic approximations, asymptotic expansions )
 30E15 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Asymptotic representations in the complex domain)
 68Q25 (Computer science :: Theory of computing :: Analysis of algorithms and problem complexity)

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