# Homework Help: Convergence of a very simple series

1. Oct 23, 2005

### Benny

Hi, there's a really simple looking series that I don't know how to deal with.
$$\sum\limits_{n = 2}^\infty {\frac{{n^2 }}{{1 - n^3 }}}$$
How would I determine whether or not this series converges by using some standard convergence tests? If a_n = (n^2)/(1-n^3) then the numerator is always positive while the denominator is always negative so that a_n is always negative. So I can't think of a way to use the comparison test, limit comparison test etc. Since a_n looks like 1/n I have a feeling that the series diverges. But I can't think of any tests to use to verify whether or not my hypothesis is correct. At first I thought about using the absolute convergence test but if a series is not absolutely convergent, it can still be convergent so that didn't really help.
Can someone help me with this one?

Note: I would like to be able to do this without the integral test if possible.

Last edited: Oct 23, 2005
2. Oct 23, 2005

### arildno

Rewrite your series S as: [itex]S=-s, s=\sum_{n=2}^{\infty}\frac{n^{2}}{n^{3}-1}[/tex]
S diverges or converges with s.

3. Oct 23, 2005

### Benny

Oh ok, thanks for the help arildno.