Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

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

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,621 Registered: 1/8/12
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?