Power series with differentiable coefficients

Question

Suppose for each $s$ in an open interval, $P_s(x)=\sum_{k=0}^\infty a_k(s) x^k$ is a power series with radius of convergence greater than R, where each $a_k(s)$ is differentiable. My question is: Is the radius of convergence of $\sum\limits_{k=0}^\infty a_k'(s) x^k$ at least R?

Answer 1

The radius of convergence can be $0$ for some $s$. Let $h \colon \mathbb{R}\to\mathbb{R}$ a continuous (or smooth, if you want your coefficients smooth) function with support in $[-1,1]$, $h(0) = 1$, $h \geqslant 0$ and $\int_{-1}^1 h(t)\,dt = 1$. Let

$$a_k(s) = k!\int_0^s h((k!)^2\cdot t)\,dt.$$

Then you have $\lvert a_k(s)\rvert \leqslant \frac{1}{k!}$, so the radius of convergence is $\infty$ for all $s$, but $a_k'(0) = k!$, so

$$\sum_{k=0}^\infty a_k'(0)x^k$$

has radius of convergence $0$.