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If $\sum a_n$ and $\sum b_n$ both converge and one of them absolutely then the Cauchy product $\sum c_n$ converges to $\sum a_n \sum b_n$. ($c_n = \sum_{k = 0}^n a_k b_{n - k}$), by Mertens Theorem.

Now, if both converge conditionally then the product does not have to converge as $a_n = b_n = (-1)^n/n$ shows. My question now is: What if $\sum a_n$ and $\sum b_n$ both converge conditionally and $\sum c_n$ converges, then is it always true that $\sum c_n$ converges to the product?

By the way, this is not homework, I'm already past the real analysis part.

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up vote 6 down vote accepted

This follows readily from Abel's convergence theorem: if $\sum_0^\infty a_n$ converges then $$\sum_0^\infty a_n=\lim_{x\to1^-}\sum_0^\infty a_n x^n.$$

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