What function does $f(x)=\sum_{k=0}^\infty k^3(x-2)^k$ represent?

Question

$$f(x)=\sum_{k=0}^\infty k^3(x-2)^k$$

I am supposed to find the $f(x)$ that this Taylor polynomial represents. How do I do this? I've tried using standard polynomials and I've tried differentiating those for $(1-x)^{-1}$ but I haven't been able to get a proper function out of it.

Answer 1

Try differentiating,

$$f(x)=\sum_{k=0}^{\infty} (x-2)^k$$

$$=\frac{1}{1-(x-2)}$$

If it converges.

Then multiply both sides by $x-2$ and repeat.