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I have been working the Rydberg equation with large $n$ to see how quantum systems when they become very large are functionally identical to classical systems. The problem I'm facing is trying to approximate the energy between two states when $n \gg 1$.

I've gotten up to the point of $$\frac{1}{n^2(1+1/n)^2} - \frac{1}{n^2}$$
I'm stuck on the first term and have seen elsewhere people approximate this to $$\frac{1}{n^2}\left(1 - \frac{2}{n}\right)$$ I'm unsure how this approximation follows. I would really like to see the steps for this in case I need to use it in the future.

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  • $\begingroup$ It is math. For small $\delta$, $1/(1-\delta) \approx 1+\delta$. And $(1+\delta)^2 \approx 1+2\delta$. The first terms in series expansions. $\endgroup$ – user137289 Nov 16 '20 at 20:38
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This is a straightforward application of the binomial approximation, which says that when $|ax| \ll 1$, $$ (1 + x)^a \approx 1 + a x. $$ In this case, we can identify $x = 1/n \ll 1$, and $a = -2$, which means that $$ \frac{1}{(1 + 1/n)^2} \approx 1 - 2 \frac{1}{n}. $$

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  • $\begingroup$ I was getting stuck on the substitution $x=1/n$. Thank you so much! $\endgroup$ – Evan Dodson Nov 17 '20 at 4:52

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