Date: Oct 6, 2017 12:24 PM
Author: Karl-Olav Nyberg
Subject: Re: Can two series, both diverges, multiplied give a series that converges?

fredag 6. oktober 2017 18.17.41 UTC+2 skrev konyberg følgende:
> fredag 6. oktober 2017 18.05.49 UTC+2 skrev Mike Terry følgende:
> > On 06/10/2017 16:21, konyberg wrote:
> > > fredag 6. oktober 2017 17.13.31 UTC+2 skrev Peter Percival følgende:
> > >> 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,

> > >>
> > >> What is the definition of the product of two infinite series?
> > >>
> > >>

> > >>> 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? Let us write the first as a series without the sigma and the
> > >>> other with sigma. s*t = (1+1/2+1/3+ ...) * t. And since the first
> > >>> from s (1 * t) diverges, how can s*t converge?
> > >>>
> > >>> KON
> > >>>

> > > It is the multiplication of the two series.
> >
> > That doesn't answer Peter's question. Each series has infinitely many
> > terms, and you need to say what you mean the product to be calculated
> > from those terms.
> >
> > If you thought this through carefully, you'd realise straight away the
> > answer to your original question, I think!
> >
> > To get you started in the right direction, suppose the first series is:
> >
> > Sum [n=1 to oo] (a_n)
> >
> > and the second is:
> >
> > Sum [n=1 to oo] (b_n)
> >
> > Now, what do you mean by the "product" of these series?
> >
> > If you feel tempted to reply "just multiply them together", then ask
> > yourself "multiply WHAT together exactly?" (Remember, multiplication is
> > an operation that takes TWO numbers, and gives a single number as the
> > answer. In the two series, you have INFINITELY many numbers...)
> >
> > Or perhaps your answer will be that the product of the two series is
> > some new third series? (If so, then say what is the n'th term of this
> > new series?)
> >
> >
> > Regards,
> > Mike.

>
> Or consider my first is sum(a) is like sum(b), where both sum(a) and sum(b) goes to inf. What is then the product of them?
> KON


Or on the other side:
One a goes to inf, the other goes to 0. What is now the product?
You can not tell if you do not know the functions defining the entities.
KON