Find the radius of convergence and the convergence at the end points of the series:
$$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$
This is what I did:
$a_n=(2+(-1)^n)^n\Rightarrow R=\frac{1}{limsup|a_n|^\frac1n}\\
for\ n=2k \to lim(2+1)=3 \\
for\ n= 2k+1 \to lim(2-1)=1 \\
limsup \ a_n^{\frac1n}=3 \Rightarrow R=\frac13$
So the series converge at $(-\frac13 , \frac13)$
But now I don't really understand what I'm being asked for the end points. Am I supposed to check if the series really converge at the two end points ?
Best Answer
Your analysis for the radius of convergence is good. To check the convergence/divergence at $x=\pm\frac13$, ask whether the terms go to $0$ as $n\to\infty$.