Algebraic Substitution: Solving F(r) Using r=r_+(1+\rho^2)

[cristo]

Homework Statement

I have this function $$F(r)=\frac{(r-r_+)(r-r_-)}{r^2}$$ and I want to make the subsitution $r=r_+(1+\rho^2)$.

Homework Equations

None.

The Attempt at a Solution

So, I sub in, to obtain $$F=\frac{[r_+(1+\rho^2)-r_-][r_+(1+\rho^2)-r_-]}{r_+^2(1+\rho^2)^2}=\frac{\rho^2[r_+(1+\rho^2)-r_-]}{r_+^2(1+\rho^2)^2}$$.

Now, the solutions say that this is equal to $$\frac{r_+\rho^2(r_+-r_-)}{r_+^2}\cdot [1+O(\rho^2)]$$, however I cannot, for the life of me, see how to get this from my line above! Can anyone help?

 

[Hootenanny]

Looks like a Taylor expansion to me, but I could be wrong...

 

[cristo]

Looks like a Taylor expansion to me, but I could be wrong...

Yup, you're correct. I never spot things like that. Thanks!