Stirling approximation revisited The Stirling approximation is useful in a
variety of differ. ent settings. The goal of the present problem is to work
through a more sophisticated treatment of this approximation than the simple
heuristic argument given in the chapter. Our task is to find useful
representations of \(n\) then since terms of the form in \(n\) arise often in
reasoning about entropy.
(a) Begin by showing that
$$n !=\int_{0}^{\infty} x^{n} \mathrm{e}^{-x} \mathrm{d} x$$
To demonstrate this, use repeated integration by parts. In particular,
demonstrate the recurrence relation
$$\int_{0}^{\infty} x^{n} e^{-x} d x=n \int_{0}^{\infty} x^{n-1} e^{-x} d x$$
and then argue that repeated application of this relation Ieads to the desired
result.
(b) Make plots of the integrand \(x^{n} e^{-x}\) for various values of \(n\) and
observe the peak width and height of this inter. grand. We are interested now
in finding the value of \(x\) for which this function is a maximum. The idea is
that we will then expand about that maximum. To carry out this step, consider
\(\ln \left(x^{\pi} e^{-x}\right)\) and find its maximum - argue why it is
acceptable to use the logarithm of the original function as a surrogate for
the function itself-that is, show that the maxima of both the function and its
logarithm are at the same \(x\). Also, argue why it might be a good idea to use
the logarithm of the integrand rather than the integrand itself as the basis
of our analysis. Call the value of \(x\) for which this function is maximized
\(x_{0}\). Now expand the logarithm about \(x_{0},\) In particular, examine
$$\ln \left|\left(x_{0}+b\right)^{n} e^{-\left(x_{0}+b\right)}\right|=n \ln
\left(x_{0}+d\right)-\left(x_{0}+8\right)$$
and expand to second order in \(\delta\), Exponentiate your result and you
should now have an approximation to the orig. inal integrand which ts good in
the neighborhood of \(x_{0}\) Plug this back into the integral (be careful with
limits of Integration) and by showing that it is acceptable to send the lower
limit of integration to \(-\infty,\) show that
$$n ! \Leftrightarrow n^{\pi} \mathrm{e}^{-n} \int_{-\infty}^{\infty}
\mathrm{e}^{-g^{2} / 2 n} \mathrm{d} \delta$$
Evaluate the integral and show that in this approximation
$$n !=n^{n} e^{-n}(2 \pi n)^{1 / 2}$$
Also, take the logarithm of this result and make an argument as to why most of
the time we can get away with dropping the \((2 \pi n)^{1 / 2}\) term.
