Exercise 17, § 3

a) Let $f$ be a vector function having derivatives of all orders on an open interval $I \subset \mathbf{R}$. Suppose that, on $I$, one has $\|f^{(n)}(x)\| \leq a n! r^n$, where $a$ and $r$ are two numbers $> 0$ and independent of $x$ and $n$; show that at each point $x_0$ the "Taylor series" with general term $(1/n!) f^{(n)}(x_0) (x - x_0)^n$ is convergent, and has sum $f(x)$ on some neighbourhood of $x_0$.

b) Conversely, if the Taylor series for $f$ at a point $x_0$ converges on a neighbourhood of $x_0$ there exist two numbers $a$ and $r$ (depending on $x_0$) such that $\|f^{(n)}(x_0)\| \leq a . n! r^n$ for every integer $n > 0$.

c) Deduce from a) and exerc. 16 b) that if, on an open interval $I \subset \mathbf{R}$, a real function $f$ is indefinitely differentiable and if there is an integer $p$ independent of $n$ such that, for all $n$, the function $f^{(n)}$ does not vanish at more than $p$ distinct points of $I$, then the Taylor series of $f$ on a neighbourhood of each point $x_0 \in I$ is convergent, and has sum $f(x)$ at every point of a neighbourhood of $x_0$.