Definition:O Notation/Big-O Notation/Sequence

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Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.

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

$\exists c \in \R: c \ge 0 : \exists n_0 \in \N : \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$

That is:

$\size {a_n} \le c \cdot \size {b_n}$

for all sufficiently large $n$.

This is denoted:

$a_n = \map \OO {b_n}$

Also defined as

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

Also denoted as

The big-$\OO$ notation, along with little-$\mathcal o$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.

In analytic number theory, sometimes Vinogradov's notations $f \ll g$ or $g \gg f$ are used to mean $f = \map \OO g$.

This can often be clearer for estimates leading to typographically complex error terms.

Some sources use an ordinary $O$:

$f = \map O g$

Also see