Date: Oct 6, 2017 11:02 AM
Author: Karl-Olav Nyberg
Subject: Re: Can two series, both diverges, multiplied give a series that converges?
fredag 6. oktober 2017 16.10.39 UTC+2 skrev Markus Klyver følgende:
> Den fredag 6 oktober 2017 kl. 15:42:22 UTC+2 skrev konyberg:
> > 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?
> > 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
> I'm not sure if the product will converge absolutely, but if multiplication is done in the naive way it'll converge.
So: limit * something = something (or limit)
Which is it?
Try it with (-1)^n and see if it is absolutely.