Exercise 9, § 4

a) Show, with the help of Stirling’s formula, that the maximum of the function $f_n(x) = \left| e^{-2x} - e^{-x} \sum_{k=0}^n (-1)^k \frac{x^k}{k!} \right|$ over 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 $\left| e^{-px} - e^{-x} g(x) \right| \leq \varepsilon$ for every $x \geq 0$ (replace $x$ by $px/2$ in a), and apply the induction hypothesis to $e^{-(p-1)x/2}$.