Re: Does this product converge ???

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1 total messages Started by Jim Roth Wed, 23 Dec 1992 11:26

#843

Author: Jim Roth
Date: Wed, 23 Dec 1992 11:26

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In article <1992Dec22.192535.29669@news.eng.convex.com>, Dave Dodson <dodson@convex.COM> writes...
>In article <1992Dec22.165148.20421@draper.com> storch@draper.com (Joel Storch) writes:

>>To prove that Product(k=3,infinity, Cos(Pi/k)) converges, first rewrite it as
>>Product(k=3,infinity,1-2 (Sin(Pi/2k))^2). This product will converge 
>>(absolutely) if the infinite series Sum(k=3,infinity,(Sin(Pi/2k))^2) 
>>converges. The Ratio Test fails here but Raabe's test shows that the series 
>>converges.

>Even simpler: The sequence of partial products is non-negative and monotone-
>decreasing. Therefore it has a greatest lower bound, which must be the limit.

But is the limit greater than zero?

Geometrically, this product arises from a nested sequence of
inscribed circles and regular n-gons (convex of course), starting
with a triangle.

- Jim

>Dave Dodson		                             dodson@convex.COM
>Convex Computer Corporation      Richardson, Texas      (214) 497-4234

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