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Topic: Can two series, both diverges, multiplied give a series that converges?
Replies: 22   Last Post: Oct 7, 2017 12:52 AM

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 Karl-Olav Nyberg Posts: 1,575 Registered: 12/6/04
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 11:21 AM

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
> >

>
>
> --
> 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

It is the multiplication of the two series.
KON