# An inverse question on Cauchy product of series

• Aug 25th 2009, 08:31 AM
zzzhhh
An inverse question on Cauchy product of series
Given two series $\displaystyle \sum\limits_{n = 0}^\infty a_n, \sum\limits_{n = 0}^\infty b_n$, and their Cauchy product $\displaystyle \sum\limits_{n = 0}^\infty c_n$ where $\displaystyle c_n=\sum\limits_{k = 0}^n a_kb_{n-k}$, if series $\displaystyle \sum c_n$ converges, dose series $\displaystyle \sum a_n$ or $\displaystyle \sum b_n$ converge too? Thanks!
• Aug 25th 2009, 06:15 PM
putnam120
No, you don't know that both $\displaystyle \sum a_n$ and $\displaystyle \sum b_n$ converge. Take $\displaystyle a_n=1$ and $\displaystyle b_n=0$.
• Aug 26th 2009, 01:20 AM
flyingsquirrel
Quote:

Originally Posted by putnam120
No, you don't know that both $\displaystyle \sum a_n$ and $\displaystyle \sum b_n$ converge.

That does not answer the question :
Quote:

Originally Posted by zzzhhh
does series $\displaystyle \sum a_n$ or $\displaystyle \sum b_n$ converge too?

• Aug 26th 2009, 03:27 AM
zzzhhh
then is it possible that both $\displaystyle \sum a_n$ and $\displaystyle \sum b_n$ diverge?
• Aug 26th 2009, 06:20 AM
putnam120
I don't think it is possible for both $\displaystyle \sum a_n,\sum b_n$ to diverge. This is clear if one of the sequences has only positive terms (or similarly only negative terms). For the case with mixed terms try violating a condition for the alternating test. I haven't worked out all the details but think that might work. I'll let you know I have figured out more.
• Aug 26th 2009, 11:40 AM
Opalg
From the wonderful book Counterexamples in analysis, an example of two (very!) divergent series whose Cauchy product is absolutely convergent. Define $\displaystyle a_0=-1$, $\displaystyle a_n=1$ for n≥1. Define $\displaystyle b_0=2$, $\displaystyle b_n=2^n$ for n≥1. Then it's easy to check that $\displaystyle c_0=-2$ but $\displaystyle c_n=0$ for n≥1. (Rock)
• Aug 27th 2009, 11:17 AM
zzzhhh
what an ingenious construction, must by Cauchy himself :-)
Thank u Opalg, as well as the great book you mentioned. thank you all!