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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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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Can two series, both diverges, multiplied give a series that
converges?

Posted: Oct 7, 2017 12:43 AM

On Fri, 6 Oct 2017, 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?

> > > 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.
> >
> > 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?

>
> There are two ways to give the product:
> 1: sum(a) * sum (b) = sum (c) or
> 2: every of sum a multiplied with the ones of sum of b.
> If the sum of any of them is not (in 2) as we like, then none of
> them are!

None of that makes any sense.