1. ## convergent or divergent?

trying to find divergence or convergence of:
the series from n=2 to infinity of (1)/((n^p)-(n^q)) (0<q<p) using limit comparison test
Using limit comparison I have

i used 1/n^p

used reciprical and multiplied by original
(1)/((n^p)-(n^q)) x (n^p/1)

simplified and ended up with this and cannot figure out what to do or if it is even right
(1)/(1-(1/n^p)) found the limit to be 1 then is it divergent by nth term test?

2. I suppose that p and q are both integers. In such case the general term $a_{n}$ of the series is...

$a_{n}= \frac {1}{n^{p}-n^{q}}$

If you apply the 'comparison test' in order to establish if the series $\sum_{n=2}^{\infty} a_{n}$ convergs or not you obtain...

$\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_{n}}= 1$

... so that the series can converge or not... you don't know!...

An easy analysis however shows that for p=1, q=0 the series diverges and in all other cases converges...

Kind regards

$\chi$ $\sigma$

3. Where are we assuming that p and q are integers?
I would do the limit comparison to $\sum {1\over n^p}$.
The limit of the ratio does go to one since 0<q<p.
So your series converges whenever p exceeds one, integer or not.