I met a problem of determining shape of a power series's value domain. The power series $f(z)=\sum a_n z^n$ is with real coefficients and $a_0=0$, $a_1>0$. A positive real number, $R$, is the series's singularity, which means $f(z)$ is divergent if $|z|>R$. Suppose $f(z)$ is convergent at $R$, I am curious what the shape of such power series $f(z)$'s value domain looks like when $|z|<R$. Could it be like a round pancake? Is it possible that there exists a $z$ in $f(z)$'s convergence domain making $f(z)>R$?
Here is an example. $\arctan z$'s Taylor series is $z-z^3/3+z^5/5-\cdots$, $1$ is its singularity. There is no $z$ in series's convergence domain making $\arctan z>1$.