Does this product converge ???

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4 total messages Started by Pavlos Kairis /ADVISOR-Gary Howe... Thu, 17 Dec 1992 18:04

#954

Author: Pavlos Kairis /ADVISOR-Gary Howe...
Date: Thu, 17 Dec 1992 18:04

13 lines
1051 bytes

Does the following product converge; if so what is the limit ?

     Product(k=3,infinity,cos(pi/k))

An upper bound of a limit appears to be pi/25.  Thanks.

-Pavlos
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Re: Does this product converge ???

#980

Author: Pierre Asselin
Date: Sat, 19 Dec 1992 19:24

16 lines
811 bytes

In <4526@winnie.fit.edu>
	rma93758@zach.fit.edu ( Pavlos Kairis /ADVISOR-Gary Howell) writes:

>Does the following product converge;
>     Product(k=3,infinity,cos(pi/k))

It converges, because...  Hmmm...  This isn't a homework, is it?

>if so what is the limit ?

For a numerical value, take the log;  directly sum a dozen terms
or so;  finish off with Euler-MacLaurin.
-- 

--Pierre Asselin, Santa Barbara, California
  pa@verano.sba.ca.us

Re: Does this product converge ???

#824

Author: Joel Storch
Date: Tue, 22 Dec 1992 16:51

9 lines
955 bytes

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.

One source of theorems on convergence of infinite Products is Whittaker & 
Watson, "A course in Modern Analysis"

Re: Does this product converge ???

#828

Author: Dave Dodson
Date: Tue, 22 Dec 1992 19:25

18 lines
1748 bytes

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.
>
>One source of theorems on convergence of infinite Products is Whittaker & 
>Watson, "A course in Modern Analysis"

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.

----------------------------------------------------------------------

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

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