find a where $\sum_{n=n(a)}^\infty (-1)^n \frac{(2+n^a)}{n}$ converges and absolutely converges

Question

Study $a$ so that the series convergences and absolute convergences I just know that $a$ is positive

I applied the ratio test, I've found that $0<a<1$

I have no idea for absolute convergence.

Thanks a lot for looking and helping.

Answer 1

Consider the absolute value $|(-1)^n \frac{2 + n^a}{n}| = \frac{2 + n^a}{n}$ for $a > 0$ we can say that $\frac{2 + n^a}{n} \ge \frac{n^a}{n} = \frac{1}{n^{1 - a}}$ and such a series is divergent hence your series doesn't converge absolutely