In my textbook, as part of an example, I am given that "

is its own Laurent expansion about z = 1, where it has a double pole."

My reasoning is this:

If we define a punctured disc centred at 1 with radius r > 0, then we can express it as a Laurent series

, where

. Having a double pole means that

, while

, for all i < -2.

How am I supposed to compute the coefficients of the Laurent series? Am I supposed to use the formula for

described above? All I seem to be able to do is the following:

, so

, which does not equal zero (what I am trying to show), and, for example,

, which does equal zero (what I am trying to show).

Can anyone shed some light on this?