Radius of convergence

Applying the ratio test gives a limit involving the absolute value of $x - a$. Setting that limit less than one produces an inequality of the form $|x - a| < R$, and $R$ is the radius.

The two extremes are real: a radius of zero means the series converges only at its centre, and an infinite radius means it converges for every real number, as the series for $e^x$ does.