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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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 Karl-Olav Nyberg Posts: 1,575 Registered: 12/6/04
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 5:42 PM

fredag 6. oktober 2017 23.24.41 UTC+2 skrev Jan følgende:
> 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

No it will not!
Sum (1/n) * Sum (1/(n+1) <> Sum (1/n * 1/(n+1))
That is the point!
KON