Mertens theorem on Cauchy products

Mertens theorem (Cauchy products): Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be convergent series (real or complex). Define the Cauchy product coefficients $c_n=\sum_{k=0}^n a_k b_{n-k}.$ If at least one of the series $\sum a_n$ or $\sum b_n$ converges absolutely , then the Cauchy product series $\sum c_n$ converges and $\sum_{n=0}^\infty c_n = \left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right).$

This result justifies multiplying power series and many other formal series manipulations when absolute convergence is present.

Proof sketch (optional): Absolute convergence lets one control double sums $\sum_{n,k}$ by comparison with $\sum |a_n|\,|b_k|$ and interchange summation order, turning the product of sums into the sum of convolutions.