Given a particular expansion, find the function in powers of x

[eclayj]

I am given this expansion:

x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + ... + k^2 x^k + ...

To get the function that yields this expansion, I cannot figure out the manipulation to (x)/(1-x)

 

[HallsofIvy]

That's probably because it is not true. To take a simple example, if x= 1/2, that sum becomes $(1/2)+ 4(1/4)+ 9(1/8)+ 16(1/16)+ ...$
which is already larger than 1/2+ 1+ 9/8+ 1= 3 and 5/8. Since every term is positive, the sum (if it converges) is larger than 3 and 5/8. But (1/2)(1- 1/2)= (1/2)/(1/2)= 1.

In fact, 1/(1- x) is the sum of the geometric series $\sum_{n=0}^\infty x^n$ so x/(1- x) is $\sum_{n=0}^\infty x^{n+1}$.