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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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 bursejan@gmail.com Posts: 5,511 Registered: 9/25/16
Re: Can two series, both diverges, multiplied give a series that converges?
Posted: Oct 6, 2017 12:51 PM

I don't know what formal multiplication John Gabriel
used. You have different options:

Option 1: Algebra Product

MATH 304: CONSTRUCTING THE REAL NUMBERS,
Peter Kahn Spring 2007
http://www.math.cornell.edu/~kahn/reals07.pdf

Option 2: Cauchy Product

https://en.wikipedia.org/wiki/Cauchy_product#Cauchy_product_of_two_infinite_series

They give different partial sums. Option 1
and Option 2 have different sweep patterns:

Option 1:

A A B _
A A B _
B B B _
_ _ _ _

Option 2:

A A B _
A B _ _
B _ _ _
_ _ _ _

Exercise: Show that although the partial sums
are differently built (different sweep pattern),
the limit is the same, if it exists.

Am Freitag, 6. Oktober 2017 18:48:08 UTC+2 schrieb burs...@gmail.com:
> Depends, try it with the definition here:
>
> MATH 304: CONSTRUCTING THE REAL NUMBERS,
> Peter Kahn Spring 2007
> http://www.math.cornell.edu/~kahn/reals07.pdf
>
> See page 12, 4.2 Algebraic operations on sequences
>
> The lecture above expounds that when the first series
> {sn} converges (by Cauchy criteria) and the second
> series {tn} converges (by Cauchy criteria) , then the
>
> result {sn}*{tn} will also converge (by Cauchy criteria).
> If one of the series diverges then this theorem of
> the lecture above is no use for you.
>
> I would say everything is possible, like:
>
> oo * 0 = 0
>
> oo * 0 = oo
>
> oo * 0 = c
>
> oo * 0 = undefined
>
> Am Freitag, 6. Oktober 2017 18:24:33 UTC+2 schrieb konyberg:

> > One a goes to inf, the other goes to 0. What is now the product?
> > You can not tell if you do not know the functions defining the entities.
> > KON