Definition:Order of Convergence

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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 $p$ if and only if 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 $1$.


First-Order Convergence

$\sequence {x_n}$ converges to $\alpha$ with order $1$ if and only if there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:

$(1): \quad \size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
$(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } {\epsilon_n} = c$

where $0 < c < 1$.


Second-Order Convergence

Definition:Order of Convergence/Second Order

Also see

  • Results about order of convergence can be found here.


Sources