Exercise 9, § 4

(a) Exercise prove, using the Stirling formula, that the maximum of the function

in the interval $[0, +\infty[$, tends to 0 with $1/n$. (b) Deduce, by induction on the integer $p$, that for every $\varepsilon > 0$, there exists a polynomial $g(x)$ such that $|e^{-px} - e^{-x}g(x)| \leq \varepsilon$ for every $x \geq 0$ (replace $x$ by $px/2$ in $a$), and use the induction hypothesis applied to $e^{-(p-1)x/2}$.