Does the Continued Product of Fractions Converge to Zero?

[SW VandeCarr]

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?

 

[Ben Niehoff]

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.

 

[SW VandeCarr]

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.

 

[AUMathTutor]

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.

 

[SW VandeCarr]

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.