Someone please help me with this seemingly geometric series
To show that the following
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!