Math Help - Someone please help me with this seemingly geometric series

Question:
To show that the following
http://www.wolframalpha.com/input/?i=sum+of+(k*(2%2F3)^k)+from+1+to+inf
is convergent conditionally or absolutely.

What I am having trouble with is the fact that it is multiplied by k. I realize that exponentiating something to k will grow faster than multiplying it by k but that is not sufficient at all for me to prove that this series converges. Actually I just used the ratio test now and found 2/3 which converges. So now a new question arises; is the ratio test valid for geometric series? Is this series a geometric one even though it is multiplied by k?

Any input would be greatly appreciated!
Thanks in advance!

Originally Posted by s3a

Question:
To show that the following
http://www.wolframalpha.com/input/?i=sum+of+(k*(2%2F3)^k)+from+1+to+inf
is convergent conditionally or absolutely.

What I am having trouble with is the fact that it is multiplied by k. I realize that exponentiating something to k will grow faster than multiplying it by k but that is not sufficient at all for me to prove that this series converges. Actually I just used the ratio test now and found 2/3 which converges. So now a new question arises; is the ratio test valid for geometric series? Is this series a geometric one even though it is multiplied by k?

Any input would be greatly appreciated!
Thanks in advance!

You may want to first derive a closed form for the series

This can be done like this



Now just plug in for x, to get .