Date: Oct 6, 2017 2:12 PM
Author: bursejan@gmail.com
Subject: Re: Can two series, both diverges, multiplied give a series that converges?

Of course you can, otherwise there would not be the
notion of "formally multiply". You can formally multiply

two infinite series, resulting in a new series. There
is a receipt as follows, namely:

Input 1: A series {s(n)}
Input 2: A series {t(n)}
Output 3: A series {r(n)}

Without the involvement of any notion of limit. The easiest
definition is to set

r(n):=s(n)*t(n).

Thus we create a new series function r, which has the value
of the first n-terms of its sum r(n), in that this

value is simply the product of the first n-terms of the
sum s(n) and and the sum t(n). If you are a programmer this

is quite simple to understand:

/* the first series */
function real s(n real) {
/* something */
}

/* the second series */
function real t(n real) {
/* something */
}

/* the product series */
function real r(n real) {
return s(n)*t(n);
}

If this were not possible, the whole construction
of real numbers from Cauchy series, as defined here,
wouldn't be so simple:

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

Because in such a construction you proceed as follows:
- You first define the *formal* multiplication
- And then you reason about whether it converges
or not

Am Freitag, 6. Oktober 2017 19:36:54 UTC+2 schrieb John Gabriel:
> You CANNOT do unless both of them converge in which case you can conclude that the product will also converge. This is a theorem. And you call yourself a mathematician? Chuckle.