```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 thenotion of "formally multiply". You can formally multiplytwo infinite series, resulting in a new series. Thereis 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 easiestdefinition is to set    r(n):=s(n)*t(n).Thus we create a new series function r, which has the valueof the first n-terms of its sum r(n), in that thisvalue is simply the product of the first n-terms of thesum s(n) and and the sum t(n). If you are a programmer thisis 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 constructionof 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.pdfBecause in such a construction you proceed as follows:   - You first define the *formal* multiplication   - And then you reason about whether it converges     or notAm 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.
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