Asymptotic Equivalence

analysis

Definition

Asymptotic Equivalence

Two sequences $(a_n{)}_{n\in \mathbb{N}}$ and $(b{)}_{n\in \mathbb{N}}$ iff:

$$a_n\sim b_n:⟺\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=1$$

Examples:

• $n^2-n\sim n^2$
• $n!\sim {\left(\frac{n}{e}\right)}^n\sqrt{2\pi n}$ (Stirling’s approximation)