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