The function $ \displaystyle\frac{1}{\sqrt{e^{x}-1}}$ doesn't have a Laurent expansion at $x=0$.

But according to Wolfram Alpha, it does have a series expansion that includes terms raised to noninteger powers. Specifically, $\displaystyle\frac{1}{\sqrt{x}}- \frac{\sqrt{x}}{4} + O(x^{\frac{3}{2}})$.

How is that series derived?

My initial thought was to use general binomial theorem. But I don't seem to get anywhere with that.