Definition:Order of Convergence/First Order
Jump to navigation
Jump to search
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)
 | This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: chapter and section number needed as applicable If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |