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Mertens theorem on Cauchy products

The Cauchy product of two convergent series converges to the product of their sums if one series converges absolutely.

Let ∑n=0∞an\sum_{n=0}^\infty a_n and ∑n=0∞bn\sum_{n=0}^\infty b_n be convergent real or complex . Define their by

cn=∑k=0nakbn−k.c_n=\sum_{k=0}^n a_kb_{n-k}.

If at least one of the two series , then ∑n=0∞cn\sum_{n=0}^\infty c_n converges and

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