# Does the serie converge or not?

• June 16th 2012, 11:52 AM
Siron
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.
• June 16th 2012, 12:04 PM
Reckoner
Re: Does the serie converge or not?
Quote:

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

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

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

$=\lim_{n\to\infty}\sqrt{\frac1{\frac1{n^2}+1}}$
• June 16th 2012, 12:31 PM
Siron
Re: Does the serie converge or not?
Thanks! ;)