Taylor expansion of 1/(r-r') for multipole expansion

[captainjack2000]

Homework Statement

Could someone please explain how the taylor expansion of 1/(r-r') turns into
( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...)

Homework Equations

The Attempt at a Solution

 

[latentcorpse]

i think it should be $\frac{1}{|\vec{r}-\vec{r'}|}$ yes?

use the 3d taylor expansion formula

$\phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{r})$ with $\phi(\vec{r}+\vec{a})=\frac{1}{|\vec{r}-\vec{r'}|}$