Write the function $\frac{1}{(z+1)(3-z)}$ as a Laurent series. $$f(z)=\frac{1}{(z+1)(3-z)}=\frac{1}{4z+4} + \frac{1}{12-4z}$$
$$\frac{1}{4z+4}=\frac{1}{4z}\frac{1}{1-\frac{-1}{z}}=\frac{1}{4z}\sum_{k=0}^{\infty} \left(\frac{-1}{z}\right)^k$$
$$\frac{1}{12-4z}=\frac{1}{12}\frac{1}{1-\frac{z}{3}}=\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{z}{3}\right)^k$$
$$f(z)=\frac{1}{4z}\sum_{k=0}^{\infty} \left(\frac{-1}{z}\right)^k+\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{z}{3}\right)^k$$
I can rewrite that as $$f(z)=\frac{1}{4z}\sum_{k=-\infty}^{0} (-1)^k z^k+\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{1}{3}\right)^k z^k$$.
I need to move the $\frac{1}{4z}$ and $\frac{1}{12}$ into the sums but finding a series that will converge to each, but I have no idea what to use for either. Any suggestions? Am I taking a wrong approach or is there an obvious series to use for this?
Edit: The center is $0$ and the region in $1 \le |z| \le 3$.
I think I can use a geometric sequence to say $\frac{1}{4z}=\sum_{k=0}^{\infty}\frac{1}{2}(\frac{1}{2})^{k-1}\frac{z}{k}$ and $\frac{1}{12}=\sum_{k=0}^{\infty}\frac{1}{36}(\frac{2}{9})^{k-1}\frac{z}{k}$. I'm pretty sure that's true, but it seems like it makes the whole thing a complicated mess. 


The function
\begin{align*}
 f(z)&=\frac{1}{(z+1)(3-z)}\\
&=\frac{1}{4(z+1)}-\frac{1}{4(z-3)}
\end{align*}
   has two simple poles at $-1$ and $3$.

Since we want to find a Laurent expansion with center $0$, we look at the poles $-1$ and $3$ and see they determine three regions.
\begin{align*}
 |z|<1,\qquad\quad
 1<|z|<3,\qquad\quad
 3<|z|
 \end{align*} 


*

*The first region $ |z|<1$ is a disc with center $0$, radius $1$ and the pole $-1$ at the boundary of the disc. In the interior of this disc all two fractions with poles $-1$ and $3$  admit a representation as power series at $z=0$.

*The second region $1<|z|<3$ is the annulus with center $0$, inner radius $1$ and outer radius $3$. Here we have a representation of the fraction with poles $-1$ as principal part of a Laurent series at $z=0$, while the fraction with pole at $3$ admits a representation as power series.

*The third region $|z|>3$ containing all points outside the disc with center $0$ and radius $3$ admits for all fractions a representation as principal part of a Laurent series at $z=0$.
A power series expansion of $\frac{1}{z+a}$ at $z=0$ is
\begin{align*}
\frac{1}{z+a}&=\frac{1}{a}\cdot\frac{1}{1+\frac{z}{a}}\\
&=\sum_{n=0}^{\infty}\frac{1}{a^{n+1}}(-z)^n
\end{align*}
The principal part of $\frac{1}{z+a}$ at $z=0$ is
\begin{align*}
\frac{1}{z+a}&=\frac{1}{z}\cdot\frac{1}{1+\frac{a}{z}}=\frac{1}{z}\sum_{n=0}^{\infty}\frac{a^n}{(-z)^n}
=-\sum_{n=0}^{\infty}\frac{a^n}{(-z)^{n+1}}\\
&=-\sum_{n=1}^{\infty}\frac{a^{n-1}}{(-z)^n}
\end{align*}

We can now obtain the Laurent expansion of $f(z)$ at $z=0$ for all three regions. 
  
  
*
  
*Region 2: $1<|z|<3$
  
  
  \begin{align*}
f(z)&=\frac{1}{4(z+1)}-\frac{1}{4(z-3)}\\
&=-\frac{1}{4}\sum_{n=1}^\infty\frac{1}{(-z)^n}-\frac{1}{4}\sum_{n=0}^\infty \frac{1}{(-3)^{n+1}}(-z)^n\\
&=\frac{1}{4}\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{z^n}+\frac{1}{4}\sum_{n=0}^\infty \frac{1}{3^{n+1}}z^n\\
\end{align*}

The Laurent expansion for the other regions can be calculated similarly.
