Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

up vote 6 down vote accepted

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.