Given this series: $$\sum_{n=1}^\infty \frac{\cos(x^n)}{n^2}, $$
Show that the series converges using the Cauchy Criterion for Series.
I attempted to solve it using the theorem, That is, let $\epsilon>0$, take M=(?) such that for all $m>n>M$, and so on...
however I ended up with $m/M^2$ and I have no idea from here how to make it less than epsilon since there is still that m in the expression.
Note that
$$\sum_{k=n+1}^m \frac1{k^2} < \sum_{k=n+1}^m \frac1{k(k-1)} = \sum_{k=n+1}^m \left(\frac1{k-1}- \frac1{k}\right) = \frac{1}{n} - \frac{1}{m} < \frac{1}{n}$$