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 $1$: $$\lim_{n\to\infty}\frac{a_n}{b_n}=1.$$

- 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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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983