Category:Definitions/First-Order Convergence

This category contains definitions related to First-Order Convergence.
Related results can be found in Category: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 $0 < c < 1$.