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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: 1,037 Registered: 5/21/05
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 5:22 PM

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