Find formula for f[x] whose expansion in powers of x is

[eclayj]

I am trying to find formula for f[x] whose expansion in powers of x is:
x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + ... + k^2 x^k + ...

I know that this is a variation on x/(1-x). I also know that x/(1-x)^2 yields expansion in power of x = kx^k. I cannot figure out the manipulation which would give the expansion k^2x^k.

Any help is appreciated

 

[LCKurtz]

I am trying to find formula for f[x] whose expansion in powers of x is:
x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + ... + k^2 x^k + ...

I know that this is a variation on x/(1-x). I also know that x/(1-x)^2 yields expansion in power of x = kx^k. I cannot figure out the manipulation which would give the expansion k^2x^k.

Any help is appreciated

Well, if I had a series expansion that gave me terms kx^k and I differentiated it that would give k²x^k-1 and if I multiplied by x ...

 

[eclayj]

Thank you very much. I need more practice and hopefully I will get better at seeing where a formula can be differentiated to get you close.