Definition:Order of Convergence

Definition
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence.

Let $\ds \lim_{n \mathop \to \infty} x_n = \alpha$.

Then $\sequence {x_n}$ converges to $\alpha$ with order $p$ there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:
 * $\size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
 * $\ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } {\epsilon_n^p} = c$ where $c > 0$

If $p = 1$, the constant $c$ is additionally required to be less than $1$.