Cauchy Product

We have two convergent series,

$ A_n = \sum^n_{k=0} a_k $ & $ B_n = \sum^n_{k=0} b_k $.

Theorem: Cauchy Product $ C_\infty = A_\infty B_\infty = \sum_{k=0} \sum_{i=0..k} a_i b_{k-i} $.

Demonstration: The product is the sum of the sums anti-diagonals of this table:

$
\begin{tabular}{ c | c | c | c | c }
(a_0 b_0) & (a_1 b_0) & (a_2 b_0) & (a_3 b_0) & a_4 b_0 & \\ \hline
(a_0 b_1) & (a_1 b_1) & (a_2 b_1) & a_3 b_1 & a_4 b_1 & \\ \hline
(a_0 b_2) & (a_1 b_2) & a_2 b_2 & a_3 b_2 & a_4 b_2 & \cdots \\ \hline
(a_0 b_3) & a_1 b_3 & a_2 b_3 & a_3 b_3 & a_4 b_3 & \\ \hline
a_0 b_4 & a_1 b_4 & a_2 b_4 & a_3 b_4 & a_4 b_4 & \\ \hline
& & \vdots & & \ddots & \\
\end{tabular}
$

Bracketed is $ C_3 $, now $A_n B_n \leq C_{2n} \leq A_{2n} B_{2n} $ so $ C_\infty $ converges. $ \qed $




Cauchy Product - Planet Math
The theorem of Mertens about the Cauchy product of infinite series - Harold P. Boas