<s>[INST] Using implicitly the extension of the binomial to real exponents, we presented the proof of the generalized binomial theorem for positive integer exponents, from which the theorem for real $k\geq 1$ can be obtained by continuity. The power series $$\sum\limits_{n=0}^\infty \frac{k(k-1)\cdots(k-n+1)}{n!}x^n $$ is known to converge for $|x|<1$ (courtesy of the ratio test). Hence for $|x|<1$ we have
$$(1+x)^k=\sum\limits_{n=0}^\infty \frac{k(k-1)\cdots(k-n+1)}{n!}x^n.$$

I'm trying to work out how this theorem applies to the sum of the infinite geometric series (consider only $x$ real). 

If comparing the infinite sum $$S=\sum\limits_{n=0}^\infty x^n$$ to our binomial expansion above, what should be $k$ to allow this simplification?[/INST]