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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 UTC7, 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



