Definition:O Notation/Big-O Notation/Sequence

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

$a_n$ is big-O of $b_n$
 * $\exists c \in \R_{\ge 0}: \exists n_0 \in \N: \forall n \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$

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 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.