# Alternating infinite sum

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?

• Ohhhhh. Okay. I thought I was going crazy, the answer key does indicate that it is not absolutely convergent. Thanks – user137720 Dec 16 '14 at 2:10