Let $$ F(x) = \sum_{n=0}^ \infty a_nx^n$$ where the power series converges in a neighborhood of the origin. Compute $$ \mu(F)= \sup \{ \delta > 0 : \text{there exist} \ \epsilon > 0 \ \text{such that} \ \int_{-\epsilon}^\epsilon |F(x)|^{- \delta}dx < \infty\}$$ where the integrals are interpreted as improper Riemann integrals if $F(0)=0$
I think we can divide this into two cases where one is when $F(0) = 0$ and when $F(0)$ is non zero. For $F(0)$ is non zero and $ 0< \epsilon<R$ where R is the radius of convergence the integral converges and using continuity we can get $ \mu (F) = \infty$
We also have to do for $F(0) = 0$
Hint: For the case $F(0)=0$ (if the function $F$ is not $0$) you can write $F(x)=x^mG(x)$ for a $G$ such that $G(0)\not =0$(and of radius of convergence $R$). Hence you can find an $\varepsilon$, $0<\varepsilon<R$, such that if $|x|\leq \varepsilon$, we have $$c=\frac{|G(0)|}{2}\leq |G(x)|\leq \frac{3|G(0)|}{2} =d$$
Now in $[-\varepsilon,\varepsilon]$, you have $$c|x|^m\leq |F(x)|\leq d|x|^m $$