Radius of Convergence

Represented as either a nonnegative real number or $\infty$ such that the series converges if $| x - a | < r$ , and diverges if $| x - a | > r$ . At $| x - a | = r$ , the series may or may not converge.

Thus,

\begin{aligned} a - r < x < a + r & power series converges \\ x < a - r and x > a + r & power series diverges \end{aligned}

Alt Definition

In other words, given a power series $\sum_{n = 0}^{\infty} c_{n} (x - x_{0})^{n}$ centred at $x_{0}$ , there exists a number R with $0 \leq R \leq \infty$ called the radius of convergence.

The series converges absolutely if $| x - x_{0} | < R$
The series diverges if $| x - x_{0} | > R$