# Definition:O Notation/Big-O Notation/Sequence

< Definition:O Notation | Big-O Notation(Redirected from Definition:Big-O Notation for Sequences)

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## Definition

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers.

**$a_n$ is big-O of $b_n$** if and only if

- $\exists c \in \R: c \ge 0 : \exists n_0 \in \N : \left({n \ge n_0 \implies \left\vert{a_n}\right\vert \le c \cdot \left\vert{b_n}\right\vert}\right)$

That is:

- $\left\vert{a_n}\right\vert \le c \cdot \left\vert{b_n}\right\vert$

for all sufficiently large $n$.

This is denoted:

- $a_n = \mathcal O \left({b_n}\right)$

## Also defined as

Some authors require that $b_n$ be nonzero for $n$ sufficiently large.

## Also denoted as

In analytic number theory, sometimes **Vinogradov's notations** $f \ll g$ or $g \gg f$ are used to mean $f = \mathcal O \left({g}\right)$.

This is clearer for estimates leading to typographically complex error terms.

Some sources use an ordinary $O$:

- $f = O \left({g}\right)$

## Also see

- Big-O Notation for Sequences Coincides with General Definition where it is shown that this definition coincides with the general definition if $\N$ is given the discrete topology.