How to write an infinite sum for $f(x)=\frac{-(x+1)}{(x-1)^3}$

Question

I am asked to evaluate the power series at $1/2$ in $(-1,1)$ for $$f(x)=\frac{-(x+1)}{(x-1)^3}$$ I get $${\frac{-x−1}{(x−1)^3}}≈12+80(x−{\frac{1}{2}})+336(x−{\frac{1}{2}})^{2}+1152(x−{\frac{1}{2}})^{3}+3520(x−{\frac{1}{2}})^4$$

My question is assuming I've done my calculations properly, how do I write this as an infinite sum?

Also when asked to evaluate a power series should I find the interval of convergence or just find the sum?

Any help would be greatly appreciated.

Thank you,

Answer 1

Note that \begin{eqnarray*} \frac{1+x}{(1-x)^3} =\sum_{n=0}^{\infty} (n+1)^2 x^n. \end{eqnarray*} So \begin{eqnarray*} \frac{1+x}{(1-x)^3} \mid_{x=\frac{1}{2}}=\color{red}{12} =\sum_{n=0}^{\infty} (n+1)^2 \left(\frac{1}{2} \right)^n. \end{eqnarray*}