Taylor series of a function

Question

I am puzzled with the following problem and I am not able to figure out the answer no matter how hard I try. Let's say we have a function $f(x)$ and we know its Taylor series is of the form $\sum_{n=0}^\infty a_nx^n$ (center $x_0=0$). We want to find the Taylor series of $f(\sqrt[k] x)$ (if it exists). We can write $f(\sqrt[k] x)= \sum_{n=0}^\infty a_nx^{n/k}$. If we differentiate $f(\sqrt[k] x)$ and find that it is not differential at $x=0$, does that mean that the Taylor series we obtained is wrong and it does not have one.