Does the Continued Product of Fractions Converge to Zero?

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Discussion Overview

The discussion centers on the convergence of the continued product of fractions of the form 1/2 x 2/3 x 3/4 x...x (n-1)/n. Participants explore whether this product converges and, if so, to what value.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions whether the continued product converges and seeks to understand its limit.
  • Another participant suggests taking the logarithm of the infinite product to transform it into an infinite sum, indicating that the convergence of the product is linked to the convergence of this sum.
  • A different participant notes that the sum of the logarithms appears to diverge to negative infinity, leading them to conclude that the limit of the continued product is zero, although they express uncertainty about the rate of convergence.
  • Some participants propose that the product converges to zero and mention that the telescoping nature of the series becomes clearer after taking the logarithm.

Areas of Agreement / Disagreement

Participants express differing views on the convergence of the continued product, with some asserting it converges to zero while others discuss the implications of the logarithmic transformation without reaching a consensus.

Contextual Notes

There are unresolved questions regarding the behavior of the logarithmic sum and its implications for the convergence of the product, as well as the nature of the convergence itself.

Who May Find This Useful

Readers interested in mathematical analysis, particularly in the study of infinite products and series, may find this discussion relevant.

SW VandeCarr
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Does the continued product of fractions 1/2 x 2/3 x 3/4 x...x (n-1)/n converge? If so, what does it converge to?
 
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The easiest way to verify is this:

1. Take the logarithm of your infinite product. This produces an infinite sum (series).

2. Use any of the well-known methods to determine if the infinite sum converges.

3. The infinite product converges if and only if the corresponding infinite sum converges.

4. If the infinite product converges, and the infinite sum converges to M, then the infinite product converges to e^M.
 
Thanks Ben Niedoff

The sum of the logarithms appears to diverge to negative infinity (-inf) but increasingly slowly. Therefore e^-inf which I took to be the limit (0) of the continued product, but the increasing slowness of convergence gave me second thoughts.
 
I think the product you gave actually converges to zero. I think it's telescoping. As was suggested, this should become more apparent after taking the logarithm.
 
AUMathTutor said:
I think the product you gave actually converges to zero. I think it's telescoping. As was suggested, this should become more apparent after taking the logarithm.

ln((n-1)/n)= ln(n-1) - ln(n) Yes, I agree. Thanks.
 

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