Can the radius of convergence of a sum of two power series be an arbitrary number?

Question

Let $\sum_{n=0}^\infty a_n z^n,\sum_{n=0}^\infty b_n z^n$ be two series with the same radius convergence $R>0$. Can the radius of convergence of their sum be any positive real number which is greater than $R$?

Answer 1

Yes, it can. If $\sum_{n=0}^\infty a_nz^n$ has radius of convergence $R\in(0,\infty)$, and if $r>R$, consider the series $\sum_{n=0}^\infty(-a_n+r^{-n})z^n$, whose radius of convergence is also $R$. But the radius of convergence of the sum of both series is $r$.