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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.
Mentioned in:
Theorems: 1
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983