Struggling to compute a power series for a complex value function

You've gone wrong because $$ \frac{1}{1-\omega}=\sum_{n\geq 0}\omega^n $$ assumes that ${|\omega|<1}$. In your case, this would mean ${|2z+4|<1}$, i.e. ${|z+2|<\frac{1}{2}}$. You have calculated the series about ${z=-2}$ with radius of convergence ${\frac{1}{2}}$.

Instead, consider $$ \frac{1}{2z+5} = \frac{1}{5}\left(\frac{1}{\frac{2}{5}z+1}\right) = \frac{1}{5}\left(\frac{1}{1-\left(\frac{-2}{5}z\right)}\right) $$ can you take it from here?

To get the expansion about $z=0$, we need to ensure $\omega$ doesn't contain an additive term and is just some scaling of $z$. By considering $5f(z)=\frac{1}{2z/5+1}=\frac{1}{1-\omega}$ and solving for $\omega$, we get $\omega =-\frac{2}{5}z$ and thus $5f(z)=\sum_{n=0}^{\infty}\frac{2^n}{5^n}(-1)^nz^n$, which gives the required answer.