Mertens' theorem

A condition ensuring the Cauchy product of two series converges to the product of their sums.

Mertens’ theorem

Mertens’ theorem: Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be series. Assume that $\sum_{n=0}^\infty a_n$ converges and that $\sum_{n=0}^\infty |b_n|$ converges. Define the Cauchy product coefficients

c_n=\sum_{k=0}^n a_k\,b_{n-k}.

Then the series $\sum_{n=0}^\infty c_n$ converges, and its sum satisfies

\sum_{n=0}^\infty c_n=\left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right).

This theorem explains when multiplication of sums is compatible with multiplication of series via Cauchy products, a key fact in manipulating power series and in results about term-by-term operations .