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Topic: Can two series, both diverges, multiplied give a series that converges?
Replies: 22   Last Post: Oct 7, 2017 12:52 AM

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Jan Bielawski

Posts: 997
Registered: 5/21/05
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 5:22 PM
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On Friday, October 6, 2017 at 6:42:22 AM UTC-7, konyberg wrote:
> Consider these two series.
> s = lim (n=1 to inf) Sum(1/n) and t = lim (n=1 to inf) Sum(1/(1+n)).
> Both series diverges, going to infinity.
> Now if we multiply these, we can argue that every product of the new series is smaller or equal to 1/n^2. So it should converge. Or can we?


The new series defined this way would be sum(1/(n(n+1))) which converges.

But if you define series multiplication this way, you won't get the
property that the result sums to the product of the two original series.
For THAT to work you need to define the product differently, see e.g.:
https://en.wikipedia.org/wiki/Cauchy_product

--
Jan


Date Subject Author
10/6/17
Read Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Markus Klyver
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
Peter Percival
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
Mike Terry
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
bursejan@gmail.com
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
bursejan@gmail.com
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
Mike Terry
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/7/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
William Elliot
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
bursejan@gmail.com
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
bursejan@gmail.com
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
bursejan@gmail.com
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Jan Bielawski
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that converges?
Karl-Olav Nyberg
10/7/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
William Elliot
10/6/17
Read Re: Can two series, both diverges, multiplied give a series that
converges?
William Elliot

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