What is the derivation of the Doppler effect for light using Taylor expansion?

[Skeptic.]

Homework Statement

This isn't strictly a homework problem, but I didn't know where else to post this. I can't get the same derivation as my lecturer for the Doppler effect of light - which is shown in the attached file. If you cannot open this, I re-wrote it further down.

Homework Equations

The Attempt at a Solution

For the part in the red box, I thought one would do this via Taylor expansion, thus I expected the $\frac{u}{c}$ to be squared, i.e f_r = f_s$(1 \pm \frac{1}{2}\frac{u}{c}^2)(1 \pm \frac{1}{2}\frac{u}{c}^2)$. I can't see why this wouldn't be the case. Could someone please tell me why I'm wrong?

(In case you cannot open the file, my lecture notes say f_r = f_s$(1 \pm \frac{u}{c})^\frac{1}{2} (1 \pm \frac{u}{c})^\frac{-1}{2}$=$(1 \pm \frac{1}{2}\frac{u}{c})(1 \pm \frac{1}{2}\frac{u}{c})$)

 

[TSny]

McClaurin series of (1+x)^1/2 = 1 + (1/2) x + ... Note that x is not squared in the second term of the right side.