Definition:Asymptotic Equality/Sequences
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Definition
Let $\sequence {a_n}$ and $\sequence {b_n}$ 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-$\oo$ notation.
Definition 3
$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:
- $a_n - b_n = \map \oo {a_n}$
where $\oo$ denotes little-$\oo$ notation.
This is denoted: $a_n \sim b_n$.