Limit Comparison Test on $\sum_{n=1}^{\infty}\frac{2n^2-1}{3n^5+2n+1}$ I have the series:
$\sum_{n=1}^{\infty}\frac{2n^2-1}{3n^5+2n+1}$
I compared it with:
$\sum_{n=1}^{\infty}\frac{n^2}{n^5}$
The limit is $\frac{2n^5-n^3}{3n^5+2n+1}$ as $n$ approaches infinity.
The limit works out to $\frac23$. I don't understand how this shows convergence? It is less than 1, so why is the p-series, the second series I have, converging?

The limit is correct. However, since the p-series will diverge if p=1,(since a p-series diverges if 0<p<=1), then the finite, positive limit shows the p-series will diverge. If the p-series diverges, then by the Limit Comparison Test, the other series must diverge as well.
