Determine whether the series is convergent or divergent. If it is convergent, find it's sum.

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

It seems to me that it should converge, because the power of the denominator is greater than the power of the numerator, but I'm having a tough time figuring out how to find the answer.