1. ## Series convergence question

This is a very simple question, but I just wanted to to make sure. The question asks to find whether or not the series [k(k+2)]/[(k+3)^3], k=1 to infinity, converges or diverges, and if it converges, to find the sum. So I used the limit method and found that the limit as k goes to inifinity is 1. The book says that the limit has to go to 0 for it to converge, so does this mean that the series diverges since the limit is 1 (if that is in fact the answer)?

2. Originally Posted by virtuoso735
This is a very simple question, but I just wanted to to make sure. The question asks to find whether or not the series [k(k+2)]/[(k+3)^3], k=1 to infinity, converges or diverges, and if it converges, to find the sum. So I used the limit method and found that the limit as k goes to inifinity is 1. The book says that the limit has to go to 0 for it to converge, so does this mean that the series diverges since the limit is 1 (if that is in fact the answer)?
$\sum{\frac{k(k+2)}{(k+3)^3}}$

note that the degree of the numerator is 2, while the degree of the denominator is 3 ... the kth term goes to 0 as k approaches infinity, but the series diverges ... do a limit comparison with the known divergent series $\sum{\frac{1}{k}}$

3. Sorry, I posted the question wrong. The denominator should be (k+3)^2, not (k+3)^3, but I guess the same thing applies.

4. much easier then ...

$\sum{\frac{k(k+2)}{(k+3)^2}}$

as k approaches infinity, kth term does not go to 0 ... series diverges.