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\title{General Mathematics and Computational Science I}
\author{Exercise 16}
\date{November 6, 2007}
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\begin{enumerate}


\item Check the conditions for Laplace's method for the function
\[
  f(x) = x-\ln x - 1
\]
which appears in the proof of  Stirling's formula.  I.e., check the
following. 
\begin{enumerate}
\item $f$ is strictly decreasing for $0<x<1$, strictly increasing for
$x>1$, and $f(1)=0$.
\item There are positive constants $b$ and $c$ such that $f(x) \geq
b\,x$ for $x \geq c$.
\item $f(x) = a \, (x-1)^2 + \psi(x) \, (x-1)^3$ where $\psi$ is a
bounded function for $x \in [1-\delta,1+\delta]$ for some $\delta >
0$.  How must $a$ be chosen?
\end{enumerate}

Hint: For (c), use either Taylor's formula with remainder (easy), or
l'H\^opital's rule for the function
\[
  g(x) = \frac{f(x)}{(x-1)^2} \,.
\]
(more elementary, but longer).



\item Use Stirling's formula to prove that
\[
  \frac{n!}{(n/2)!^2} \sim 2^n \, \sqrt{\frac{2}{\pi n}} \,.
\]

\end{enumerate}
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