Write the Laurent series for the function $f(z)=\frac{1}{1+z}$ $1<|z|<\infty$
I did $$\frac{1}{1-z}=\sum_{i=0}^\infty z^n\rightarrow \frac{1}{1+z}=\sum_{i=0}^\infty (-1)^nz^n$$ Is it right? There is a general procedure for me to check if the series I wrote is correct for a particular function?
In this case the restriction $1<|z|<\infty$ is just by the fact of function have a singularity point in 1?
The problem is that the series you wrote converges only if $|z|<1$. $$ \frac 1 {1-z} = \frac{1/z}{(1/z)-1} = \frac {-1} z \left(\frac 1 {1-\frac 1 z}\right) = \frac{-1} z \left( 1 + \frac 1 z + \frac 1 {z^2} + \frac 1{z^3} +\cdots \right) =\cdots $$ This converges when $|z|>1$.