complex analysis - Laurent series expansion of $f(z)=frac{1}{(z-1)^2(z+1)^2}$

Let $f(z)=\frac{1}{(z-1)^2(z+1)^2}$. While trying to expand this function into the Laurent series, convergent in $P(0,1,2):=\lbrace z\in\mathbb{C}:1<|z|<2\rbrace$, a few questions popped into my mind.

1. We can write $f(z)=\frac{1}{4}\left(\frac{1}{(z-1)^2}+\frac{1}{(z+1)^2}\right)$. Both functions inside parentheses are complex derivatives of functions which have immediate Laurent series expansion: $\frac{1}{1-z}$ and $-\frac{1}{z+1}$. Now, can we differentiate the obtained series term by term to get the desired expansion of $f$? If so, is it because the Laurent series is convergent almost uniformly?



2. Could someone verify that the Laurent series of $f$ is convergent in $P(0,1,\infty)$?