Suppose that $f \in C^{\infty} (-\infty , \infty)$ and that $$\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0$$ for all $a\in \mathbb{R}$. Prove that $f$ is analytic on $(-\infty , \infty)$ and $$f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}x^k, \hspace{1cm} x\in \mathbb{R}$$
I have the theorem: If $f \in C^{\infty}(a,b)$ and $f^{(n)}(x) \geq 0$ for all $x \in (a,b)$ and $n\in \mathbb{N}$, the $f$ is analytic on $(-\infty ,\infty)$ and $$f(x) = f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!}x^k$$
If I can show that $\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0 \Rightarrow f^{(n)}(x) \geq 0$ then I think I would apply this theorem and be done, but I am not sure how to do that. Any help would be appreciated, or if this isn't the correct way to go about this a nudge in the right direction would be nice as well.