[Math] Convergence Test $\sum_{n=2}^\infty\frac{1}{n^p-n^q}$

Question

To test the convergence of the series $\sum_{n=2}^\infty\frac{1}{n^p-n^q}$ I tried the limit comparison test. $0<q<p.$ My $a_n=n^p-n^q$ and my $b_n=\frac{1}{n^p-n^q}$. The limit comparison test says:

If $$\lim_{n\to\infty}\frac{b_n}{a_n}=1$$ then $\sum a_n$ converges iff $\sum b_n$ converges.

Now, since the $\lim_{n\to\infty}a_n=\infty \ne0$, $\sum a_n$ can not converge. And if $\sum a_n$ does not converge then $\sum b_n$ can not converge. After my try I discovered that I'm wrong according to the solutions. Why and where am I wrong?

Thank you.

Answer 1

Hint: Wrong $a_n$. It should be $\frac{1}{n^p}$. Then use standard facts about $p$-series.