TITLE: Radius of convergence of a sum of power series
QUESTION [13 upvotes]: I have  two series
$\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$ 
$\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$
with radius of convergence $2$ and $3$ respectively.
How can I find the radius of convergence for the following?

$\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}$

Can anyone help me?

REPLY [4 votes]: Since 
$\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}=\displaystyle\sum_{n=1}^{\infty} a_n x^{n}+\displaystyle\sum_{n=1}^{\infty} b_n x^{n} $ 
 it is clear that the left hand side converges for values of $x$ exactly when both of the series on the RHS converge.
Since this means that the LHS will converge on the intersection of the discs of convergence on the RHS, you know the radius of convergence for the LHS is the same as the smaller radius on the RHS.

I only have confidence in this answer for absolutely convergent series. Perhaps its necessary after all to resort to a test for convergence, rather than relying on this naive arithmetic.