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I come across a expansion of the partition fct. of a simple harmonic oscillator $q$ as:

$$q=x^{-1}(1-\frac{x^2}{24}+...) \tag{1}$$

where $x=h\nu/kT$. It’s easy to get $$q=\frac{e^{-x/2}}{1-e^{-x}}=\frac{1}{e^{x/2}-e^{-x/2}}.$$ Expand the RHS, I got:

$$x^{-1}(1+x^2/24+...)^{-1}. \tag{2}$$

But how can I go from (2) to (1)?

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When $z$ is small $(1+z)^{\alpha}=1+\alpha z+{\cal O}(z^{2})$. (This is just the leading term in Newton's Binomial Theorem. So the leading term in $x^{-1}(1+x^{2}/4!+\cdots)^{-1}$ is $x^{-1}(1-x^{2}/4!+\cdots)$. This is obtained by setting $\alpha=-1$ and $z=x^{2}/4!+\cdots$. This approximation omits terms of ${\cal O}(z^{2})$ and higher, but those are all of higher order in $x$ as well.

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