Convergence Speed of a Sequence

Topics: Numerical Analysis - Sequence - Limit of a Sequence

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Let:

• $\{{\beta }_n{\}}_{n=1}^{\infty }$ be a sequence that converges to $0$
• $\{{\alpha }_n{\}}_{n=1}^{\infty }$ be a sequence that converges to $\alpha \in \mathbb{R}$

We say that $\{{\alpha }_n\}$ converges to $\alpha$ with a convergence speed of $O({\beta }_n)$ if:

$$|{\alpha }_n-\alpha |\le K|{\beta }_n|$$

…where:

• $n$ is sufficiently big
• ${\beta }_n\ne 0$ for every $n$
• $K$ is a constant that’s independent from $n$

We denote this with:

$${\alpha }_n=\alpha +O({\beta }_n)$$