\frac{-1/2}{x+1}=\frac{-1/2}{(x-1)+2}=\frac{-1}4\,\frac 1{1+(x-1)/2}=-\frac 14\sum_{n=0}^\infty\frac{(-1)^n}{2^n}(x-1)^n=\sum_{n=0}^\infty\frac{(-1)^{n+1}}{2^{n+2}}(x-1)^n