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CrosisBH

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- 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)

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

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

**binomial expansion***yields*

[tex] \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)[/tex]

*Thus*

[tex]\frac{1}{𝓇^2_{+}} - \frac{1}{𝓇^2_{-}} \cong \frac{d}{r^2}\cos\theta[/tex]

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

[tex](a+b)^2 = a^2 + 2ab + b^2[/tex]

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.