What test should I use to classify this series?

Question

Classify $$\sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}$$ as absolutely convergent, conditionally convergent or divergent.

I am not sure whether I should use integral test or comparison test.

Do you know what's the best test to classify this series?

Thanks in advance.

Answer 1

Note:

$$\left | \sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}\right | \le \sum_{n=1}^\infty \frac{|x^2+\cos^n{x}|}{n^2} \le \sum_{n=1}^\infty \frac{ x^2+1}{n^2} = (x^2+1) \frac{\pi^2}{6}$$

Therefore the sum is absolutely convergent for any finite value of $x$ by the comparison test. Thanks to @N.S. and @vadim123 for correcting and clarifying the reasoning here.