Power Series

In the form

\sum_{n = 0}^{\infty} C_{n} (x - a)^{n}

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 $| x - x_{0} | > R$

Random Things of Note

In the case of $x = a$ , the power series will always converge
Generally, using the Ratio Test or Root Test will be most successful