Thread: Does the serie converge or not?

1. Does the serie converge or not?

Hi,

I'm not able to check if the following serie converges or not:
$\sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$

The ratio test didn't work, and I think I need a majorant or a minorant but I can't find one.

2. Re: Does the serie converge or not?

Originally Posted by Siron
Hi,

I'm not able to check if the following serie converges or not:
$\sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$
Use limit comparison with the convergent p-series $\sum_n\frac1{n^{3/2}}$.

Spoiler:
$\lim_{n\to\infty}\frac{1/\sqrt{n+n^3}}{1/n^{3/2}}$

$=\lim_{n\to\infty}\frac{n^{3/2}}{\sqrt{n+n^3}}$

$=\lim_{n\to\infty}\sqrt{\frac{n^3}{n+n^3}}$

$=\lim_{n\to\infty}\sqrt{\frac1{\frac1{n^2}+1}}$

Thanks!