Let $f$ be an infinitely differentiable function on an interval $I$. If $a \in I$ and there are positive constants $C$, $R$ such that for every $x$ in a neighborhood of $a$ and every $k$ it holds that
$|f^{(k)}(x)| \leq C \frac{k!}{R^k}$
then prove that the Taylor series of $f$ about $a$ converges to $f(x)$.
I think a good approach would be to estimate the error term. I'm not sure how to proceed exactly though. Thoughts?