If a positive series converges in square sum, will its average series converge in square sum?

Question

If a positive series converges in square sum, will its average series converge in square sum?

Specifically, assume $ s _ n > 0 $ for all $ n \in \mathbb Z _ { > 0 } $ and $ \sum _ { n = 1 } ^ \infty s _ n ^ 2 < + \infty $. Let $ \overline S _ n = \frac { \sum _ { i = 1 } ^ n s _ i } n $ for each $ n \in \mathbb Z _ { > 0 } $. Is it true that $ \sum _ { n = 1 } ^ \infty \overline S _ n ^ 2 < + \infty $?

Answer 1

This is a direct consequence of Hardy's inequality for $ p = 2 $, which states $$ \sum _ { i = 1 } ^ \infty \overline S _ n ^ 2 \le 4 \sum _ { i = 1 } ^ \infty s _ n ^ 2 \text . $$