Rudin Theorem 3.26 Series of Nonnegative Terms

Question

3.26 Theorem If $0\le{x}<1$ then $$ \sum_{n=0}^\infty x^n = \frac{1}{1-x}.$$ If $x \ge 1 $ the series diverges.

Why does Rudin allow $0 \le x$ instead of $0 < x$? The result is not true for $x=0$ (even if $n$ is indexed at 1 to avoid $0^0$).

Answer 1

The result is true for $x=0$ if using the not uncommon convention that $0^0=1$. There is an extensive discussion of this on Wikipedia https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero .