Can you tell me what you think about my solution to this problem? In case it's wrong, or needs changes, just something like "try looking at this", "consider that"... a hint is enough, please no complete answers.
Let $R_1$ be the radius of convergence of $\sum a_n z^n$, and $R_2$ the radius of convergence of $\sum b_n z^n$. What can you say about the convergence radius of $\sum (a_n \pm b_n)z^n$?
My work so far:
Assume that $R_1<R_2$.
If $z<R_1$, $\sum a_n z^n$ and $\sum b_n z^n$ converge so $\sum (a_n \pm b_n)z^n$ will also converge.
If $R_1 < z < R_2$ then $\sum a_n z^n$ will diverge and $\sum b_n z^n$ will converge therefore $\sum (a_n \pm b_n)z^n$ will diverge.
If $R_1 < R_2 < z$, both $\sum a_n z^n$ and $\sum b_n z^n$ diverge, so at first I was unsure about what happened in this case, but since all power series have a convergence radius $R$ such that they converge for $z<R$ and diverge for $R<z$, $R_1$ must be the convergence radius for $\sum (a_n \pm b_n)z^n$.