# Approximating $\sum_{j=0}^{\infty} e^{-wj} = 1/w +1/2 +w/12 -w^3 /720 +w^5/30240 +…$

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.

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 $u \to 0$.