Suppose that $R \gt 0$ and $f : (-R, R) \to \R$ is infinitely differentiable. Suppose further that for any $0 \lt r \lt R$ there is a sequence $(M_{r,k})_k$ such that for all $x \in (-r, r)$, $$\abs{f^{(k)}(x)} \leq M_{r,k} \qquad\qquad\text{and}\qquad\qquad \lim_{k\to\infty} M_{r,k}\frac{r^k}{k!} = 0.$$ Show that for every $x \in (-\color{red}{R}, \color{red}{R})$, $$\int_0^x f(t)\,dt = \sum_{k=1}^\infty f^{(k-1)}(0)\frac{x^k}{k!}.$$
(Hint: first show that our assumptions bounding the derivatives of $f$ mean that the Taylor polynomials converge uniformly to $f$ in the sense of this 142A assignment; next show that if a sequence of functions $(f_n)$ converges uniformly to $f$ then the sequence of integrals $\int_a^b f_n$ converges to $\int_a^b f$; then put these two statements together.)