What you want to do is treat x as a constant and do the ratio test.
So for we get
= .
Now for this to converge you want when k goes to infinity. So when k does go to infinity so we are left with
hence .
ive run into the following question and i really don't know where to start. i havnt come across any questions involving 2 variables yet.
For what values of x does the following series converge?
k=1(sigma)infinity = x^k/((4^k)*k)
any help would be great.
First improve your notation, more readable would be:
sum( x^k/(k 4^k) , k=1 .. infty)
better yet learn to use the math type setting facility on this site, see the tutorial here
The general term of your series is:
when this is bounded in absolute value by which is the general term of a convergent geometric series and hence your series is convergent.
when the series is the harmonic series and hence divergent, when the series is the alternating harmonic series and hence convergent.
When , does not go to zero as goes to infinity hence the series diverges.
CB