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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?

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    The way I stated it is correct.2012-03-08

2 Answers 2