I don't know, maybe it's late and I just need to knock off, but I can't quite nail the power series for the function $\displaystyle \frac{4}{x-1}$ with the center at zero. $\displaystyle \frac{1}{x-1}$ is $\displaystyle (-1)^nx^n$, however, so could I simply put the 4 out in front as a coefficient and use that? $\displaystyle 4(-1)^nx^n$ perhaps?