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

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

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?