Definition:Order of Convergence

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

Let $\alpha \in \R$.

Let $p \in \R_{\ge 1}$.

Then $\sequence {x_n}$ converges to $\alpha$ with order $p$ there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:
 * $(1): \quad \size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
 * $(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } { {\epsilon_n}^p} = c$ where $c > 0$

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

Also see

 * Order of Convergence Implies Convergence