Interval of convergence

Definition

Let $\sum a_n (x - c)^n$ be a real power series. Its interval of convergence is the set of real numbers $x$ for which the series converges.

Remarks

Let $R$ be the radius of convergence. The interval of convergence contains $(c - R, c + R)$ and lies in $[c - R, c + R]$. Each endpoint must be checked on its own. For $\sum x^n / n$, the interval is $[-1, 1)$. For $\sum x^n$ it is $(-1, 1)$, and for $\sum x^n / n^2$ it is $[-1, 1]$.

Explanations

L1 human

The radius of convergence fixes the interval up to its two endpoints. At each endpoint the series may converge or diverge, so the interval may be open, closed, or half open.