Radius and interval of convergence

A power series converges within a distance $R$ of its center; the endpoints need separate checking.

A power series' radius of convergence $R$ (found via the Ratio or Root Test) is the distance from the center $a$ within which the series is guaranteed to converge: it converges for $|x-a|<R$ and diverges for $|x-a|>R$ . The exact behavior AT the two endpoints $x=a\pm R$ is NOT determined by $R$ alone and must be checked separately (each endpoint could converge or diverge independently), giving the full interval of convergence.

Examples

$\sum\dfrac{x^n}{n}$ has radius $R=1$ (Ratio Test); testing endpoints separately shows $x=-1$ converges (alternating harmonic series) but $x=1$ diverges (harmonic series), so the interval is $[-1,1)$ .