# Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem.

If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$.

How to prove it?

• Your title does not correspond to your question. – Chris Eagle Oct 6 '12 at 13:33
• Please consider edit your post according to what you want to ask – leo Oct 6 '12 at 15:19

Hint: Cesàro summable means that $\lim_{n\to\infty}\frac{c_1+\cdots+c_n}n$ exists. Note that $\frac{c_1+\cdots+c_n}n = \frac{c_1+\cdots+c_{n-1}}{n-1}\cdot(1-\frac1n)+\frac{c_n}n$.