Let $\sum _ { n \geqslant 0 } a _ { n }$ and $\sum _ { n \geqslant 0 } b _ { n }$ be two series of complex numbers, convergent with respective sums $A$ and $B$. We denote $\left( c _ { n } \right) _ { n \in \mathbb { N } }$ the sequence with general term $c _ { n } = \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k }$ and $\left( C _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of partial sums associated defined by $C _ { n } = \sum _ { k = 0 } ^ { n } c _ { k }$. Prove that $$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } C _ { k } \right) = A B \qquad \text{(Cauchy)}$$
Let $\sum _ { n \geqslant 0 } a _ { n }$ and $\sum _ { n \geqslant 0 } b _ { n }$ be two series of complex numbers, convergent with respective sums $A$ and $B$. We denote $\left( c _ { n } \right) _ { n \in \mathbb { N } }$ the sequence with general term $c _ { n } = \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k }$ and $\left( C _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of partial sums associated defined by $C _ { n } = \sum _ { k = 0 } ^ { n } c _ { k }$. Prove that
$$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } C _ { k } \right) = A B \qquad \text{(Cauchy)}$$