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Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ two (not necessarily absolute) convergent series. We denote their Cauchy product by

$\sum_{n=0}^\infty c_n$

where $c_n=\sum_{l=0}^n a_lb_{k-l}$.

Assume that $\sum_{n=0}^\infty c_n$ converges. Is it true that in this case we have that

$(\sum_{n=0}^\infty a_n)\cdot (\sum_{n=0}^\infty b_n)=\sum_{n=0}^\infty c_n$.

Can this be seen "easily" (in an elementary way)?

zylri
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  • This is a direct result of Cesaro's theorem which says that convergence of the individual series implies the cauchy product is Cesaro summable with Cesaro sum equal to the product of the two series. Now, if a series is convergent, their Cesaro and typical sums are identical which gives you the result you desire. I'm not sure if you would call this elementary, but it is "easy" to see if you know the results. – David Mar 07 '17 at 14:18
  • We can also prove it using Abel's theorem. That might not count as elementary either, however. – Daniel Fischer Mar 07 '17 at 21:42

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