Mertens' Convergence Theorem

Theorem
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be real or complex sequences.

Let:
 * $\ds \sum_{n \mathop = 0}^\infty a_n$ converge to $A$


 * $\ds \sum_{n \mathop = 0}^\infty b_n$ converge to $B$.

Let either $\ds \sum_{n \mathop = 0}^\infty a_n$ or $\ds \sum_{n \mathop = 0}^\infty b_n$ converge absolutely.

Then:
 * $\ds \paren {\sum_{n \mathop = 0}^\infty a_n} \paren {\sum_{n \mathop = 0}^\infty b_n} = \sum_n \paren {\sum_{j \mathop + k \mathop = n} a_j b_k}$

If both $\ds \sum_{n \mathop = 0}^\infty a_n$ and $\ds \sum_{n \mathop = 0}^\infty b_n$ converge absolutely, then so also does the Cauchy product.