Consider a sequence $(a_n)$ satisfying $\lim_{n\to\infty} a_n = a$. Let $b_n = \frac{1}{n} \sum_{i=1}^n a_i$. I have already known that $\lim_{n\to\infty} b_n = a$. I am wondering is there any estimation about the convergence speed of $(b_n)$. Specifically, do we have some equality such as: $ \lim_{n \to \infty} \frac{a_n - a}{b_n-a} = 1 $ or if there exists some constant $C$ such $|a_n-a| \geq (\text{ or }\leq) \ C|b_n -a|$ for large enough $n$?
2026-02-23 10:47:00.1771843620
Convergence speed of Cesaro mean
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