Often in math we estimate quantities when we can't find the exact values.
For power series, it seems that we (almost) always find the exact value of the radius of convergence.
I am interested in power series examples where we can show that the radius of convergence $R$ verifies $a<R<b$ for some $a,b >0$, but for which we can't easily find the exact value of $R$.
You probably will not be interested in this example. If so, I will delete it.
Let$$a_n=\begin{cases}1&\text{ if both $n$ and $n+2$ are prime}\\2^{-n}&\text{ otherwise}\end{cases}$$and let $R$ be radius of convergence of the power series $\sum_{n=0}^\infty a_nx^n$. Then $1\leqslant R\leqslant2$. Actually, $R=1$ or $R=2$. But it is not known which of these assertions hold.