Definition:O Notation/Big-O Notation/Sequence

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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