I've been asked to show that if $f$ is in $C^{\infty}$, and there exist $L>0$ such that $\vert f^{(n)}(x)\vert \leq L$, then for any $x_0,x$, $$f(x)=\sum_{n=0}^{\infty} f^{n}(x_0)\frac{(x-x_0)^n}{n!}$$
I'm not sure of how to prove this, initially I though that the fact that the function is infinitely differentiable implied the existence and equality of the series expansion, however we saw some examples like $h(x)=e^{\frac{-1}{x^2}}$ when $x\neq 0$ and $0$ if $x=0$ where this was not the case.
I appreciante any insights on this problem.