I have the following infinite sum: $$ \sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}} $$ Because there is a $(-1)^n$ I deduce that it is a alternating series. Therefore I use the alternating series test: $$ \lim_{n\to\infty} \frac{1}{\sqrt{n}} $$ Because this limit is decreasing and approaching $0$ I thought it should therefore be convergent. However in the answer key it uses a different method to get a different answer. It instead takes the absolute value of the series:
$\left|\frac{(-1)^n}{\sqrt{n}}\right| = \frac{1}{\sqrt{n}}$ and says because this is a divergent p series that the series is divergent.
Why is the alternating series test not applied here?