I want to show
\begin{equation} 1 + (\phi z)^{-1} + (\phi z)^{-2} + … \end{equation}
converges to $\frac{1}{1- (\phi z)^{-1}}$ when $|\phi| > 1$ and $z \geq 1$.
What I tried is
\begin{align*} 1 + (\phi z)^{-1} + (\phi z)^{-2} + … & = \lim_{n \to \infty} \sum_{i=0}^{n} (\phi z)^{-i}\\ & = \lim_{n \to \infty} \frac{1 - (\phi z)^{-(n+1)}}{ 1- (\phi z)^{-1}} \end{align*}
I can see that $(\phi z)^{-(n+1)}$ converges to zero when $|\phi| > 1$ and $z \geq 1$ and thus the result is verified. However, what would happen when $|\phi| \geq 1$ and $z > 1$. To me it seems still converges.
Edit: I am trying to prove this from a book.
However, I don't really see why this condition has to be this way. Why can't it be $|\phi| \geq 1$ and $|z| > 1$. Aren't they equivalent?
You are right, the whole question can be handled with $w:=(\phi z)^{-1}$ such that $|w|<1$ (which is compatible with $|\phi|>1,|z|\ge1$).
Then
$$\left|\sum_{k=0}^nw^k\right|\le\sum_{k=0}^n\left|w\right|^k<\frac1{1-|w|}.$$