Definition:Little-O Notation/Sequence/Definition 2

Definition

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

Let $b_n \ne 0$ for all $n$.

$a_n$ is little-$\oo$ of $b_n$ if and only if:

$\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = 0$

The expression $\map f n \in \map \oo {\map g n}$ is read as:

$\map f n$ is little-$\oo$ of $\map g n$

Similarly, when expressed in the notation of sequences, $a_n \in \map \oo {b_n}$ is read as:

$a_n$ is little-$\oo$ of $b_n$

While it is correct and accurate to write:

$\map f n \in \map \oo {\map g n}$

or:

$a_n \in \map \oo {b_n}$

it is a common abuse of notation to write:

$\map f n = \map \oo {\map g n}$

or:

$a_n = \map \oo {b_n}$

This notation offers some advantages.

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