Let $\sum a_nz^n$ and $\sum b_nz^n$ power series with radiuses of convergences $R_1,R_2$ respectively. Suppose the radius of convergence of $(\sum a_n+b_n)z^n$ is $R$. Find an example in which $\infty>R>\min \{R_1,R_2\}$, given $R_1=R_2$.
What am I to do here? Previously I was required to show that $R=\{R_1,R_2\}$ (Given $R_1\ne R_2$, how does it matter?) and now that? Besides, $(\sum a_n+b_n)z^n=\sum a_nz^n+\sum b_nz^n$ and if $R$ is bigger the the minimum then one of the series must not converge, isn't that so? I would appreciate your help.
Best Answer
Consider the series for $\frac1{1-x}$ and the series for $\frac1{2-x}-\frac1{1-x}$. Their sum is the series for $\frac1{2-x}$.