Represent f(x)=1/(1+x) as a power series around x=1

Question

There is a question similar to this here: Represent $ f(x) = 1/x $ as a power series around $ x = 1 $

I’m hoping to solve this in a similar manner (not using a Taylor series).

My main issue is that I am having trouble coming up with a way to rewrite 1/(1+x) in a way that will allow the series to be centered at 1.

Answer 1

$$\dfrac1{1+x}=\dfrac1{2+(x-1)}=\dfrac12\dfrac1{1+\dfrac12(x-1)}$$

$$=\dfrac12\left(1-\dfrac12(x-1)+\dfrac1{2^2}(x-1)^2-...\right)$$

$$=\dfrac12-\dfrac14(x-1)+\dfrac18(x-1)^2-...$$

Answer 2

You could use the binomial theorem extension to negative powers