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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



