Infinite product, arranging of terms

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Discussion Overview

The discussion revolves around the concept of infinite products formed from a set of integers and their inverses, specifically examining how the arrangement of terms affects the outcome of the product. Participants explore two different arrangements of terms and question the implications of these arrangements on convergence and existence of the product.

Discussion Character

  • Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant proposes two arrangements for forming an infinite product: one where integers p are paired with inverses 1/q such that q>p, leading to a product that appears to converge to 0, and another where 1/q is paired with p>q, suggesting a product greater than 1.
  • Another participant questions whether the outcome of an infinite product can depend on the order of multiplication.
  • A different participant expresses skepticism about the validity of infinite products or sums depending on term arrangement, suggesting that such dependence undermines their existence.
  • One participant references the property of conditionally convergent sums being rearrangeable to converge to any value, relating it to the discussion of infinite products.
  • Another participant asserts that for an infinite product to converge, the terms must approach 1, which they argue is not the case in the initial example presented.

Areas of Agreement / Disagreement

Participants express differing views on the dependence of infinite products on the arrangement of terms, with some suggesting that it is possible while others challenge this notion. The discussion remains unresolved regarding the implications of these arrangements on convergence and the existence of the product.

Contextual Notes

Participants highlight potential issues with the assumptions underlying the formation of infinite products, particularly regarding the convergence criteria and the implications of rearranging terms.

Ookke
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Consider a set that has integers p and inverses 1/q, infinitely many of both. We are about to form a product of all numbers in this set, with two different arrangements.

Arrange terms, way 1:
For each p (countably many), select 1/q so that q>p, the resulting product term is (p * 1/q) < 1. The supply of p's are exhausted this way, and in addition there are infinitely many left-over 1/q's. As far as I understand (and possibly not), the product of all should be 0.

Arrange terms, way 2:
For each 1/q select p>q, the resulting terms are (1/q * p) > 1 with infinitely many left-over p's. This seems an infinite product.

Missing something? Thanks in advance.
 
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Doesn't it seem possible that such an "infinite product" will depend on the order in which the products is done?
 
HallsofIvy said:
Doesn't it seem possible that such an "infinite product" will depend on the order in which the products is done?

I find it hard to believe that the product (or sum of a series) would depend on arranging of terms. It's easier to think that product or sum does not exist in that case.

Also this kind of examples cast some doubt over our ability to choose infinitely many items from a set, though this is somewhat common in mathematics. There is a big difference between hand-picked set and a set that is assumed to exist by some axiom.
 
Any conditionally convergent sum can be rearranged to converge to any given value. That was why I asked.
 
For a product to converge, the terms have to go to 1. This is not the case in your example.
 

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