How is a binomial expansion done?

[CrosisBH]

TL;DR

Can someone give me a basic high level overview on how to do a binomial expansion?

Summary: Can someone give me a basic high level overview on how to do a binomial expansion?

I'm studying for my E&M test and going over multipole expansion. I'm particularly confused about these lines (Griffiths E&M 4th Edition)

$$𝓇^2_{\pm} = r^2 \left(1\mp \frac{d}{r} \cos\theta + \frac{d^2}{4r^2}\right)$$

We're interested in the régime r>>d, so the third term in negligible, and the binomial expansion yields

$$\frac{1}{𝓇^2_{\pm}} \cong \frac{1}{r} \left( 1 \mp \frac{d}{r}\cos\theta\right)^{-1/2}\cong \frac{1}{r}\left(1\pm\frac{d}{2r}\cos\theta\right)$$

Thus

$$\frac{1}{𝓇^2_{+}} - \frac{1}{𝓇^2_{-}} \cong \frac{d}{r^2}\cos\theta$$

I understand how the first line was derived, and I understand the first half on the second line, but I don't understand how the approximation was made in the second half. It's called a binomial expansion apparently, but all my research seems to point toward expanding an integer power binomial

$$(a+b)^2 = a^2 + 2ab + b^2$$

And anything about a generalized form is written with binomial coefficients which I can't seem to wrap my head around, and right now it seems beyond my math level to understand it formally. Could someone give me a physics level rigor on how this expansion is done? This'll probably be on my next exam and I want to understand it.

 

[jedishrfu]

The general binomial expansion is:

$(1 + x)^n = 1 + nx + \frac {n(n - 1)} { 2! } x^2 + ...$

It's still valid if n = -1/2

and so in your case $x = \frac d { 2r } cos \theta$

and so you get $\frac 1 {r_{-}^2} = \frac 1 r (1 - \frac d { 2r } cos \theta)$

and $\frac 1 {r_{+}^2} =\frac 1 r (1 + \frac d { 2r } cos \theta)$

and hence you get

$\frac 1 {r_{+}^2} - \frac 1 {r_{-}^2} = \frac 1 r (1 + \frac d { 2r } cos \theta) - \frac 1 r (1 - \frac d { 2r } cos \theta) = \frac d { r^2 } cos \theta$

Does that make sense ?

https://socratic.org/questions/how-do-you-use-the-binomial-series-to-expand-1-x-1-2

 

[mfb]

It is also a special case of the Taylor expansion, expand $(1+x)^c$ around x=0.

 

[mathwonk]

this famous result is due already to isaac Newton:

https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html

 

[CrosisBH]

That makes perfect sense. After I posted this, I consulted my current Math Professor (Diff eq and Linear Algebra) and he explained it was an infinite series but he forgot the exact equation but I should be able to find it. Then I come here and jedishrfu posted the equation I needed and went through the math, and mfb explained it was derived from the Taylor Series. I honestly haven't touched a Taylor Series since Calc 2 a year ago and I forgot they were a thing. My professor would probably give a very similar problem to this one and I should just memorize.
$$(1+x)^{-1/2} \cong 1-\frac{1}{2}x$$

I've heard that Taylor Series expansions are so common in physics so I should just start getting used to them. Thank you everyone!