Power series properties. The sum of two series. 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. 

$$1+x+\frac{x^2}{2^2}+x^3 + \frac{x^4}{2^4} + x^5 + \frac{x^6}{2^6}+\cdots $$
and 
$$1-x+\frac{x^2}{2^2}-x^3 + \frac{x^4}{2^4} - x^5 + \frac{x^6}{2^6}- \cdots$$
both have radius of convergence $1.$ But the sum of these series is
$$2+ \frac{x^2}{2} + \frac{x^4}{2^3}+ \frac{x^6}{2^5}+\cdots ,$$
which has radius of convergence $2.$
