Definition:Order of Convergence/First Order

Definition

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

Let $\alpha \in \R$.

$\sequence {x_n}$ 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 $0 < c < 1$.

Also known as

First-order convergence is also known as linear convergence.

Also see

• Results about first-order convergence can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): first-order convergence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 12. (of convergence of a sequence)

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• 1997: Walter Gautschi: Numerical Analysis: An Introduction