Cauchy product - Knowlpedia

A Cauchy product of two series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ is the series $\sum_{n=0}^\infty c_n$ where $c_n=\sum_{k=0}^n a_k\,b_{n-k}$ for each $n\ge 0$ .

Convergence conditions

Under suitable hypotheses, the Cauchy product represents multiplication of sums; for instance, if both series are absolutely convergent then the Cauchy product converges and sums to the product of the two sums, and Mertens' theorem gives a common sufficient condition beyond absolute convergence. Cauchy products are especially natural when multiplying power series.

Examples

If $a_n=b_n=r^n$ with $|r|<1$ , then $c_n=\sum_{k=0}^n r^k r^{n-k}=(n+1)r^n$ , so $\left(\sum_{n=0}^\infty r^n\right)^2=\sum_{n=0}^\infty (n+1)r^n$ .
If $a_n=b_n=1$ for all $n$ , then $c_n=n+1$ , so the Cauchy product is $\sum_{n=0}^\infty (n+1)$ (a divergent series).