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!