So I want to find the radius of convergence of $\frac{z}{1-z-z^2}:=\sum_{n=0}^{\infty}F_nz^n$ and also show that $F_n$ are the Fibonacci number.
$\underline{\text{My attempt}}$
To show that $F_n$ are the Fibonacci numbers is it enough to see that
\begin{align*}
\frac{z}{1-z-z^2}&=z(1-(z+z^2))^{-1}\\
&=z+z^2+2z^3+3z^4+5z^5+…
\end{align*}
where the coefficients seems to be the Fibonacci numbers?
To find the radius of convergence it is easy to see that $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta =\frac{1-\sqrt{5}}{2}$ are simple poles of $\frac{z}{1-z-z^2}$. So we can write $$\frac{z}{1-z-z^2}=\frac{z}{\sqrt{5}}\left( \frac{1}{z+\alpha}-\frac{1}{z+\beta}\right)$$however I'm not quite sure how to continue. Any help would be appreciated!
Best Answer
$$1-z-z^2=1+\dfrac14-\left(z+\dfrac12\right)^2=\dfrac54\left(1-\left(\dfrac{2z+1}{\sqrt5}\right)^2\right)$$
So we need $$\left|\dfrac{2z+1}{\sqrt5}\right|<1$$