Expanding Fraction with Infinite Value: Simple Expansion Homework Solution

[spacetimedude]

Homework Statement

I am trying to expand $\frac{1}{(1+\frac{a^2}{z^2})^{1/2}}$ for z>>a.

Homework Equations

The Attempt at a Solution

First, I rearranged the equation to $(1+\frac{a^2}{z^2})^{-1/2}$. After this, since z>>a, can I assume z takes a value of infinity and say the first term is 1+0=1? And I am not sure what to do for the second term. I take the first derivative which is $-\frac{1}{2}(1+\frac{a^2}{z^2})^{-3/2}(\frac{-2a^2}{z^3})$ and not sure what to do with it.
Any help will be appreciated.

 

[Ray Vickson]

Homework Statement

I am trying to expand $\frac{1}{(1+\frac{a^2}{z^2})^{1/2}}$ for z>>a.

Homework Equations

The Attempt at a Solution

First, I rearranged the equation to $(1+\frac{a^2}{z^2})^{-1/2}$. After this, since z>>a, can I assume z takes a value of infinity and say the first term is 1+0=1? And I am not sure what to do for the second term. I take the first derivative which is $-\frac{1}{2}(1+\frac{a^2}{z^2})^{-3/2}(\frac{-2a^2}{z^3})$ and not sure what to do with it.
Any help will be appreciated.

So, are you not just trying to expand $(1 + x^2)^{-1/2}$ for small $x = a/z$?

 

[spacetimedude]

So, are you not just trying to expand $(1 + x^2)^{-1/2}$ for small $x = a/z$?

I'm having difficulty understanding how to expand for small x. I've only come across questions that ask something like "expand this function around x= some number". Do I take x=0?

EDIT: Ah, so do I take x=0 and is the second non-zero term the term using second derivative?