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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




Re: Can two series, both diverges, multiplied give a series that converges?
Posted:
Oct 7, 2017 12:52 AM


On Fri, 6 Oct 2017, 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
> No it will not! > Sum (1/n) * Sum (1/(n+1) <> Sum (1/n * 1/(n+1)) > That is the point!
Big deal. So divergent series act differently than convergent series. Are you demanding otherwise?



