Category:Definitions/Order of Convergence

This category contains definitions related to Order of Convergence.
Related results can be found in Category:Order of Convergence.

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence of numbers or vectors in a normed vector space.

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 $c > 0$