# The convergence interval of two series's sum

I have the following statement:

If $\sum_{n=0}^{\infty}a_{n}x^n$ converges for $|x| < R_{1}$ and $\sum_{n=0}^{\infty}b_{n}x^n$ converges for $|x| < R_{2}$ , then $\sum_{n=0}^{\infty}(a_{n}+b_{n})x^n$ converges for $|x| < R_{1}+R_{2}$ as well.

I have tried to disprove this statement for a long time but couldnt get to a final answer. Thanks alot!

• – complexmanifold Jun 1 '16 at 10:54
• To find a counterexample, try letting both series be the same (say, $a_n=b_n=1$ for all $n$ for example). – Matthew Towers Jun 1 '16 at 10:56

Take $a_n = 0$ for every $n$ then $R_1 = +\infty$ and $b_n$ such that $R_2 <\infty$ (ex. $b_n = 1$ so $R_2 = 1$). So $R_1 +R_2 = +\infty$ which is a contradiction to the fact that
$\sum (a_n+b_n)x^n = \sum b_n x^n$ so $R_1 + R_2 = R_2 = +\infty.$