Question

How do you find the Taylor series of `f(x)=1/x` ?

Answer 1

The Taylor series of `f(x)=1/x` centered at `1` is

`f(x)=sum_{n=0}^infty(-1)^n(x-1)^n`.

Let us look at some details.

We know

`1/{1-x}=sum_{n=0}^infty x^n`,

by replacing `x` by `1-x`

`Rightarrow 1/{1-(1-x)}=sum_{n=0}^infty(1-x)^n`

by rewriting a bit,

`Rightarrow 1/x=sum_{n=0}^infty(-1)^n(x-1)^n`

I hope that this was helpful.