```Date: Oct 6, 2017 12:17 PM
Author: Karl-Olav Nyberg
Subject: Re: Can two series, both diverges, multiplied give a series that converges?

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
```