Date: Oct 6, 2017 11:32 AM
Author: bursejan@gmail.com
Subject: Re: Can two series, both diverges, multiplied give a series that converges?

For two series {sn} and {tn} just define
the product series as {sn*tn}, note this is NOT:

sum_i^oo ai*bi /* NOT the product */

where sn=sum_i^n ai and tn=sum_i^n bi. The
above is also not the Cauchy product.

Here is the product of two harmonic series:

sn sn^2
1 1
1.5 2.25
1.833333333 3.361111111
2.083333333 4.340277778
2.283333333 5.213611111
2.45 6.0025
2.592857143 6.722908163
etc...

It pretty much diverges, since sn < sn^2,
and sn diverges. And its not sum 1/n^2.

Am Freitag, 6. Oktober 2017 17:13:31 UTC+2 schrieb Peter Percival:
> 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
> >

>
>
> --
> Do, as a concession to my poor wits, Lord Darlington, just explain
> to me what you really mean.
> I think I had better not, Duchess. Nowadays to be intelligible is
> to be found out. -- Oscar Wilde, Lady Windermere's Fan