What kind of series is the function $\frac{x}{e^x-1}$?

How to expand the series $\dfrac{1}{e^x-1}$

Series expansion for $f(x)=\frac{x}{e^x-1}$

In these topics, I have asked for a way to develop the function $\dfrac{x}{e^x-1}$ into series. I won't do that here. From what I see, this function should not have a Maclaurin series (centered at 0) since the function and its higher derivative remain undefined when x=0. Then what is the name for the series derived from the topics that I mention above. Is it a Laurent series or Puisseux series?

Solution 1:

The function itself is not a series, of any kind. A series is one way to represent a function.

I think the term you're looking for is Laurent series. This is the generalisation of Maclaurin series that allows for negative integer exponents.

Puisseux series allow for fractional exponents as well. That is not what these examples use, so while the term may technically be applicable (I don't know much about them personally), it's not what I would call it.