I am given the following power series: $$\sum_{n=1}^\infty\frac{z^{n}+z^{-n}}{n^2}$$ My attempt: $$\sum_{n=1}^\infty\frac{z^{n}+z^{-n}}{n^2}=\sum_{n=1}^\infty\frac{z^{2n}+1}{n^2z^{2n}}\cdot z^n$$
I then tried to solve for the radius using $z_n$ but can't find a solution. Can someone explain me how to get the radius of convergence and then how to calculate the points to which this series converges? Thanks a lot.
If $|z|>1$, then $$\left|{z^n+z^{-n}\over n^2}\right|\ge\left|{z^n\over n^2}\right|-\left|{z^{-n}\over n^2}\right|>1$$ for large $n$ (since $|z^n/n^2|\to\infty$ and $z^{-n}/n^2\to0$ as $n\to\infty$), so the series diverges (a necessary condition for $\sum a_n$ to converge is $a_n\to0$ as $n\to\infty$).
If $|z|<1$, then $$\left|{z^n+z^{-n}\over n^2}\right|\ge\left|{z^{-n}\over n^2}\right|-\left|{z^n\over n^2}\right|>1$$ for large $n$ (since $|z^{-n}/n^2|\to\infty$ and $z^n/n^2\to0$ as $n\to\infty$), so the series diverges.
If $|z|=1$, then the series converges, by comparison to $\sum2n^{-2}$.