Cesàro summability implies convergence of (conventional) sum when $\sum_{n=1}^\infty na_n^2<\infty$

Question

Suppose that series $\sum_{n=1}^\infty na_n^2<\infty$ and $\sum_{n=1}^\infty a_n$ is Cesàro summable. How do you show that $\sum_{n=1}^\infty a_n$ converges? I know that if $\lim_{n\rightarrow \infty} na_n=0$ then the result is true, is this any useful in here?

thanks

Answer 1

Write $S_N = \sum_{n=1}^N a_n$. Then, $$ \left| S_N - \frac{1}{N} \sum_{n=1}^N S_n \right|^2 = \left|\sum_{n=1}^N \frac{n-1}{N} a_n \right|^2.$$ This is less than $$\sum_{n=1}^N \frac{(n-1)^2}{N} |a_n|^2$$ using Cauchy's inequality, which is then bounded by $\sum_{n=1}^N n |a_n|^2$, giving the convergence.