Starting with the geometric series 1+x+x^2+x^3+... (-1<x<1), find the power series for x/[(1-x)^2] in powers of x.
Where is the series valid?
Find the sum ∑(1,∞) n/(2^n).
That is equal to the derivative of the power series I showed previously. So you have a power series for the above, now all you need to do is account for an x in the numerator and you will have an answer.
From my last post, the above is equal to . Now multiply this by x. x represents a constant thus can be included inside the summation. Simplification leads to the power series of the expression from your problem.