Deriving taylor series for v/c and gamma

[warfreak131]

Homework Statement

The velocity of a proton relative to
our galaxy is vp/c = 1-(0.5*10^20), i.e. almost one. Such protons are actually observed.

When velocity it very nearly one $$\gamma$$ is very large. 1/$$\gamma$$
is very small. Use Taylor series to show that for v almost one we have

v/c$$\approx$$1-(1/2)(1/$$\gamma^{2}$$)...

Homework Equations

The Attempt at a Solution

I haven't done Taylor series for about a year now, and I don't quite remember how to do it. The teacher gives us some basic info on the mathematics needed to do the homework, but I can't seem to figure out a relationship. I've also looked for explanations of taylor series relevant to my question, but I haven't found anything

 

[vela]

First, express v/c as a function of 1/γ². Then expand that function as a Taylor series about 1/γ²=0.

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...

 

[Mindscrape]

I imagine that you could also just do a taylor expansion on gamma and it would give you similar results, though I have to admit I haven't looked into it. This is just what's normally done.