Infinite product of consecutive odd/even, even/odd ratios

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Discussion Overview

The discussion revolves around the infinite product of ratios of consecutive integers, specifically the expression \(\prod_{n=1}^\infty(n/(n+1))\). Participants explore the limit of a finite product as \(N\) approaches infinity, examining its behavior and implications.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant poses the question regarding the infinite product \(\prod_{n=1}^\infty(n/(n+1))\).
  • Another participant defines \(P(N)=\prod_{n=1}^N\frac{n}{n+1}\) and suggests evaluating the limit as \(N\) approaches infinity.
  • A subsequent reply calculates the limit of \(P(N)\) and concludes it approaches 0 as \(N\) tends to infinity.
  • A later response expresses gratitude for the clarification provided by the previous participant.

Areas of Agreement / Disagreement

There is a general agreement on the approach to evaluating the limit of the product, but the implications of the result may not be fully explored or agreed upon.

Contextual Notes

The discussion does not address potential assumptions or the broader implications of the limit, nor does it explore alternative interpretations of the infinite product.

Loren Booda
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?=\prod_{n=1}^\infty(n/(n+1))
 
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Let
P(N)=\prod_{n=1}^N\frac{n}{n+1}.

Then you're looking for \lim_{n\to\infty}P(n).

P(1) = 1/2, P(2) = 1/2 * 2/3 = 1/3, P(3) = ... do you see what's going on here? What's the limit going to be?
 
\lim_{N \to \infty}P(N) = \lim_{N \to \infty} \frac{1}{N+1} = 0.
 
Thank you, CR. I just needed a little shaking up.
 

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