Definition:Big-O Notation/Sequence

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

The statement


 * $a_n = \mathcal O \left(b_n\right)$

is equivalent to:
 * $\displaystyle \exists c \in \R: c \ge 0 : \exists n_0 \in \N : (n \geq n_0 \implies |a_n| \leq c \cdot |b_n|)$

That is, $|a_n| \leq c \cdot |b_n|$ for all sufficiently large $n$.

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