# Definition:Asymptotically Equal/Sequences

## Definition

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$.

### Definition 1

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

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$\ds \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = 1$

### Definition 2

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$a_n - b_n = \map o {b_n}$

where $o$ denotes little-o notation.

### Definition 3

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$a_n - b_n = \map o {a_n}$

where $o$ denotes little-o notation.

This is denoted: $a_n \sim b_n$.