# Approximation of Rydberg Equation at Very Large $n$

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.

• 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. – user137289 Nov 16 '20 at 20:38

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 $$\frac{1}{(1 + 1/n)^2} \approx 1 - 2 \frac{1}{n}.$$
• I was getting stuck on the substitution $x=1/n$. Thank you so much! – Evan Dodson Nov 17 '20 at 4:52