Definition

Let ${r_{n}} \in (0, \infty)$ be the rate of convergence, and ${x_{n}}$ be a Sequence, then

$x_{n} = o (r_{n}) \Leftrightarrow n \to \infty lim \frac{x _{n}}{r _{n}} = 0 \Leftrightarrow \forall ϵ > 0, \exists N \in N, \forall n \geq N, \frac{x _{n}}{r _{n}} < ϵ$

$x_{n} = O (R_{n}) \Leftrightarrow n \to \infty lim sup \frac{x _{n}}{r _{n}} < \infty \Leftrightarrow \exists M \in R^{+}, \exists N \in N, \forall n \geq N, \frac{x _{n}}{r _{n}} \leq M$