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So I know that $$\sum_{j=0}^{\infty} e^{-wj} = \frac{1}{1-e^{-w}},\quad \text{if}\quad e^{-w} <1,$$ and that $$e^{-w} = 1-w+w^2/2 -w^3/6 +w^4/24,$$ but don't see a way to combine the two results to give the approximation on the title.

Any help is appreciated, thanks.

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Hint. One has $$ \begin{align} \frac{1}{1-e^{-w}}&=\frac1{w-w^2/2+w^3/6-w^4/24} \\\\&=\frac1{w}\cdot\frac1{1-w/2+w^2/6-w^3/24} \end{align} $$ then one may set $u=-w/2+w^2/6-w^3/24 $ and, as $w \to 0$, one may use, $$ \frac1{1+u}=1-u+u^2-u^3+u^5+O(u^6) $$ as $u \to 0$.

One gets the announced result.

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