I'm taking a introdutory course on complex analysis and after seeing theorems of Taylor and Laurent series for complex functions I got to the exercises and found this one that I'm struggling to understand what is actually asked:
Develop the function $z^{-1}$ in powers of $(z-1)$. Then obtain by derivation the developing of $z^{-2}$ in powers of $(z-1)$. Give the region of validity.
My thoughts were: I must write the Taylor series of $z^{-1}$ around $z_0 = 1$, then the region where the series converge are the interior of the circle $|z-1| = 1$. Then I'm allowed to derivate inside this region and get the representation of $z^{-2}$ in powers of $(z-1)$, and the region of validity will be the disk $|z-1| < 1$.
But thinking on Laurent series, I think I can actually write the same expression and the region where the series representation holds will be any annular region $1 < |z-1| < \infty$.
So the answer for the region of validity of the power series representation will be the union of these two regions? Thanks in advance.