Let $\{a_n \}_{n \geq 1}$ and $\{b_n \}_{n \geq 1}$ be two sequences of real numbers such that the infinite series $\sum\limits_{n=1}^{\infty} a_n$ and $\sum\limits_{n=1}^{\infty} b_n$ are both convergent in the Cesaro sense i.e. \begin{align*} \lim\limits_{n \to \infty} \frac 1 n \sum\limits_{k=1}^{n} s_k & < + \infty \\ \lim\limits_{n \to \infty} \frac 1 n \sum\limits_{k=1}^{n} t_k & < + \infty \end{align*}
where $\{s_k \}_{k \geq 1}$ and $\{t_k\}_{k \geq 1}$ are sequences of partial sums of the series $\sum\limits_{n=1}^{\infty} a_n$ and $\sum\limits_{n=1}^{\infty} b_n$ respectively. Can I say that $\sum\limits_{n=1}^{\infty} a_n b_n$ is convergent in the Cesaro sense? If "yes" then what can I say about it's limit in terms of the limits of the given two series?
Best Answer
No. Consider $a_{n}=b_n=(-1)^n$. Then both of them are Cesaro summable but $c_n=a_n \cdot b_n= 1$ isn't, since $\lim\limits_{n \to \infty} \frac 1 n \sum\limits_{k=1}^{n} u_k= \lim\limits_{n\to \infty}\frac{1}{n}\frac{(n+1)n}{2}=\infty$