Definition:Order of Convergence/Second 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 $2$ if and only if there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:

$\size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$

$(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } { {\epsilon_n}^2} = c$

where $0 < c < 1$.

Also known as

Second-order convergence is also known as quadratic convergence.

Also see

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

Sources

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