Knowlpedia

Mertens' theorem

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

Mertens' theorem: Let ∑n=0∞an\sum_{n=0}^\infty a_n and ∑n=0∞bn\sum_{n=0}^\infty b_n be series. Assume that ∑n=0∞an\sum_{n=0}^\infty a_n converges and that ∑n=0∞∣bn∣\sum_{n=0}^\infty |b_n| converges. Define the coefficients

cn=∑k=0nak bn−k.c_n=\sum_{k=0}^n a_k\,b_{n-k}.

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

∑n=0∞cn=(∑n=0∞an)(∑n=0∞bn).\sum_{n=0}^\infty c_n=\left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right).
Remarks

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