Does this series converge conditionally $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$

Question

$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$

According to my understanding, if $\sum\left|a_n\right|$ diverges but $\sum a_n$ converges, then the series is conditionally convergent.

For $\sum\left|a_n\right|$ my series can be test via the p-series test and since $\frac{1}{10} \lt 1$ it diverges.

So next I test $\sum a_n$ using the alternating series test and find that it is a decreasing series and the limit converges to 0.

Thus, I came to the conclusion that this is conditionally convergent. Is this correct?

Answer 1

Yes, it is. (Note however that to be fully correct, the statement you make should read that "$(\lvert a_n\rvert)_n$ is a decreasing sequence," not that "$\sum_n a_n$ is a decreasing series.")

Answer 2

A monotonically descending alternating series converges, is that series a A monotonically descending alternating series ?