```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?> KONOr 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
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