# Determining convergence or divergence

• March 5th 2010, 12:15 PM
Determining convergence or divergence
The problem asks whether this sum will converge or diverge:

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

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

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

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

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

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

Is it possible to simply take the limit of this or just even say that because the denominator is raised to a power of $\frac {1} {2}$ 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.
• March 5th 2010, 12:26 PM
Krizalid
it diverges, we can bound the general term as follows: $\frac{n^{2}+n}{\sqrt{n^{3}+2n^{2}}}>\frac{n^{2}}{\ sqrt{2n^{3}+2n^{3}}}=\frac{\sqrt{n}}{2},$
• March 5th 2010, 07:26 PM
Miss
The limit comparison test with $\sum \frac{1}{n}$ also works.