Find out whether series diverges or converges

Question

I have this math problem: "determine whether the series converges absolutely, converges conditionally, or diverges."

I can use any method I'd like. This is the series:$$\sum_{n=1}^{\infty}(-1)^n\frac{1}{n\sqrt{n+10}}$$

I though about using a comparison test. But I'm not sure what series I can compare $\sum_{n=1}^{\infty}\frac{1}{n\sqrt{n+10}}$ to.

Answer 1

Converges absolutely

$|u_n|=\frac{1}{n\sqrt{(n+10)}}<\frac{1}{n}.\frac{1}{n^{\frac{1}{2}}}=\frac{1}{n^{\frac{3}{2}}}$

Answer 2

We can use Leibniz theorem: if a_n is a monotone sequence that converges to 0, then the series $\sum_{n=1}^{\infty}(-1)^n a_n$ converges. Let´s have $a_n=\dfrac{1}{n\sqrt{n+10}}$. So we have to prove that $lim$ an=0 and that it is monotone. If you can show that, you have that it converges