Theorem.
If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$.
How to prove it?
2026-05-13 19:43:31.1778701411
Cesaro summable implies that $c_{n}/n$ goes to $0$
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Hint: Cesàro summable means that $\lim_{n\to\infty}\frac{c_1+\cdots+c_n}n$ exists. Note that $\frac{c_1+\cdots+c_n}n = \frac{c_1+\cdots+c_{n-1}}{n-1}\cdot(1-\frac1n)+\frac{c_n}n$.