Cauchy product

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 $n\ge 0$.

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).