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## Definition: Asymptotical Approximation

Two real sequences $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N},$ $a_n,b_n\neq 0,$ are called asymptotically equivalent, notated by $(a_n)_{n\in\mathbb N}\sim (b_n)_{n\in\mathbb N},$ if the following limit equals $1$: $$\lim_{n\to\infty}\frac{a_n}{b_n}=1.$$

### Notes

• In this definition, the sequences $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ do not have to be convergent themselves.
• In general, the sequence of the differences $(a_n-b_n)_{n\in\mathbb N}$ does not have to converge, either.
• If $(a_n)_{n\in\mathbb N}\sim(b_n)_{n\in\mathbb N},$ we say also that $(a_n)_{n\in\mathbb N}$ can be asymptotically approximated by $(b_n)_{n\in\mathbb N}$ and vice versa.

| | | | | created: 2019-09-10 21:03:15 | modified: 2019-09-10 21:10:38 | by: bookofproofs | references: [581]

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