Prod[1-(n+1)^3, n=1,...,Inf]
Eric W. Weisstein
eww at wolfram.com
Tue Apr 18 16:00:01 CEST 2006
On Mon, 17 Apr 2006, Dean Hickerson wrote:
> Zak Seidov wrote:
>
>> Prod[1-(n+1)^3,n=1,...,Inf]=
>> Cosh[(Sqrt[3]*Pi)/2]/(3*Pi)=
>> 0.809396597366290109578680478726.
>
> The exponent should be -3, not 3. (And it could be simplified to
> prod[1-n^-3, n=2,...,inf].)
>
>> Is this cons known?
The product is very similar to
http://mathworld.wolfram.com/InfiniteProduct.html eqn (23) and might very
well have a similar formula for arbitrary power. Any takers?
Cheers,
-E
> Not to me. But Mathematica is able to evaluate the infinite product in
> closed form in less than a second, so presumably it's not hard to prove.
> Mma also gives closed forms for the product with the exponent replaced
> by other negative integers, in terms of trig, hyperbolic trig, and gamma
> functions.
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