Given two series , and their Cauchy product where , if series converges, dose series or converge too? Thanks!

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- August 25th 2009, 09:31 AMzzzhhhAn inverse question on Cauchy product of series
Given two series , and their Cauchy product where , if series converges, dose series or converge too? Thanks!

- August 25th 2009, 07:15 PMputnam120
No, you don't know that both and converge. Take and .

- August 26th 2009, 02:20 AMflyingsquirrel
- August 26th 2009, 04:27 AMzzzhhh
then is it possible that both and diverge?

- August 26th 2009, 07:20 AMputnam120
I don't think it is possible for both 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.

- August 26th 2009, 12:40 PMOpalg
From the wonderful book Counterexamples in analysis, an example of two (very!) divergent series whose Cauchy product is absolutely convergent. Define , for n≥1. Define , for n≥1. Then it's easy to check that but for n≥1. (Rock)

- August 27th 2009, 12:17 PMzzzhhh
what an ingenious construction, must by Cauchy himself :-)

Thank u Opalg, as well as the great book you mentioned. thank you all!