Develop $f(z) = \dfrac{1}{z(1+z)}$ into a Laurent series for the region $0< |z| < 1$

Question

Here is the solution to the problem (I need some clarification):

After partial fraction decomposition we have $\dfrac{1}{z(1+z)} = \dfrac{1}{z} - \dfrac{1}{1+z}$

But since we're only interested by the region $0< |z| < 1$, we can rewrite $\dfrac{1}{1+z} = \dfrac{1}{1-(-z)} = \sum_{k≥0} (-z)^k$

Therefore $f(z) = \dfrac{1}{z} -  \sum_{k≥0} (-z)^k$ is the Laurent series.

The problem I have is that we defined a Laurent series as a series of the form $\sum_{k=-\infty}^{\infty}a_k (z-c)^k$ And $\dfrac{1}{z} -  \sum_{k≥0} (-z)^k$ is clearly not of that form.

Answer 1

Yes, it is of that form. With $a_j = 0$ for $j < -1$, and $a_k = (-1)^{k+1}$ for $k \geq -1$. And, of course, $c = 0$.

Answer 2

Note that $\frac{1}{z(z+1)}$ has a simple pole at $z=0$ so the Laurent's expansion has $a_{-n}=0$ for all $n\neq 1$, $n\in \mathbb N$