Prove that if the function $f$ is infinitely differentiable on an interval $(r, s)$ containing $a$, then for any $x \in(r, s)$ and any positive integer $n$ we may expand $f(x)$ in the form

$$f(a)+(x-a) f^{\prime}(a)+\frac{(x-a)^{2}}{2 !} f^{\prime \prime}(a)+\cdots+\frac{(x-a)^{n}}{n !} f^{(n)}(a)+R_{n}(f, a, x),$$

where the remainder term $R_{n}(f, a, x)$ should be specified explicitly in terms of $f^{(n+1)}$.

Let $p(t)$ be a nonzero polynomial in $t$, and let $f$ be the real function defined by

$$f(x)=p\left(\frac{1}{x}\right) \exp \left(-\frac{1}{x^{2}}\right) \quad(x \neq 0), \quad f(0)=0 .$$

Show that $f$ is differentiable everywhere and that

$$f^{\prime}(x)=q\left(\frac{1}{x}\right) \exp \left(-\frac{1}{x^{2}}\right) \quad(x \neq 0), \quad f^{\prime}(0)=0,$$

where $q(t)=2 t^{3} p(t)-t^{2} p^{\prime}(t)$. Deduce that $f$ is infinitely differentiable, but that there exist arbitrarily small values of $x$ for which the remainder term $R_{n}(f, 0, x)$ in the Taylor expansion of $f$ about 0 does not tend to 0 as $n \rightarrow \infty$.