# Thread: Determining convergence or divergence

1. ## Determining convergence or divergence

The problem asks whether this sum will converge or diverge:

$\displaystyle \sum_{n=1}^{\infty} \frac {n(n+1)} {\sqrt {n^3+2n^2)}}$

What I had done from this point was to attempt to simplify the sum, by factoring the denominator so that instead of having:

$\displaystyle \sqrt {n^3+2n^2)}$

$\displaystyle \sqrt {n^2(n+2)}$ which becomes: $\displaystyle n\sqrt {(n+2)}$

So then back to the initial fraction, the factor of n would cancel and what would be left is:

$\displaystyle \frac {(n+1)} {\sqrt {n+2)}}$

Is it possible to simply take the limit of this or just even say that because the denominator is raised to a power of $\displaystyle \frac {1} {2}$ that the series must diverge? I am tempted to simply just note the power value and say that the sum must diverge then.

Thanks for the help, it's much appreciated.

2. it diverges, we can bound the general term as follows: $\displaystyle \frac{n^{2}+n}{\sqrt{n^{3}+2n^{2}}}>\frac{n^{2}}{\ sqrt{2n^{3}+2n^{3}}}=\frac{\sqrt{n}}{2},$

3. The limit comparison test with $\displaystyle \sum \frac{1}{n}$ also works.