Ok, I did a substitution and got
Does that look right?
Thanks, I edited my post, I had 2 typos. The powers of two were supposed to be n in the series. I'll look over your working and see why it's different.
EDIT: It seems that your answer is equivalent! Cool
Well, the condition that is satisfied by for which the original function has a pole. Hence you cannot expect to get a Laurent-Series expansion that converges for all such values of .
But it might be that one of the two series that you have already found converges for some values other than that satisfy .
Let me put it this way: you can never have an essential singularity inside the disc of convergence of a power series, or inside the annulus of convergence of a Laurent series.
But you can (and typically do have such a singularity) right on the boundrary of that disc / that annulus of convergence. Stronger: what limits the size of the disc / annulus of convergence is the existence of such a singularity on its boundary.
Convergence or divergence of those power / Laurent series expansions for points that lie right on the boundary of the disc / annulus of convergence is a separate problem that might need some further trickery to decide.