It is defined in a note that
$\frac{1}{1-\phi x} = 1 + \phi x + (\phi x )^2 + … $
provided that $|\phi| < 1$ and $|x| \leq 1$.
Why it is necessarity to have $|x| \leq 1$ instead of $|x| < 1$.
Because if I let $z = \phi x$, this becomes a geometric series with $|z| < 1$. So $|ax| < 1$ implying that
(I) $|a| < 1$ and $|x| < 1$
(II) $|a| \leq 1$ and $|x| < 1$
(III) $|a| < 1$ and $|x| \leq 1$
Why do we pick only the last condition?
Edited:
I found this also in a time series book (Time series analysis and its applications)
There are many more possibilties ! For example $a=1/3$ and $x=2$, ....
Your question was: "Why do we pick only the last condition?"
My answer: " Ask the author of the note."