[seqfan] Re: Another approximation of pi

[israel at math.ubc.ca]

Of course there's also 

pi = ((-1)^(n+1) 2 Zeta(2 n) (2 n)! /bernoulli(2 n))^(1/(2 n))/2

so

pi ~= ((-1)^(n+1) 2 (1 + 2^{-2n} + ...) (2 n)! /bernoulli(2 n))^(1/(2 n))/2

Robert Israel
University of British Columbia

On Oct 3 2012, Peter Luschny wrote:

>Alonso del Arte:
>> As far as I know, our own Daniel Forgues is the first to notice that 
>> sqrt(9.87654321) = 3.1426968... which is about as good an approximation 
>> of pi as 22/7. For some Sequences of the Day in September, to suggest 
>> that a keyword:cons sequence ought to be chosen, I put in the number 
>> 9.87654321 purely as a placeholder. To my pleasant surprise, Dan added 
>> his observation to the September 30 entry.
>
>Well, the problem here is that you did not stop your placeholder at 9.87.
>This would have given Daniel the chance to notice a much better
>approximation of Pi than 22/7.
>
>Alexander R. Povolotsky:
>>Also it might be worth noting that
>>7901234568/987654321*123456789=~987654312
>>and as a result
>>(79.01234568*1.23456789)^1/4
>>gives as well
>>3.142696798...
>
>Now this gives me the chance to point to my marvelous formula
>
> \pi= \left({\frac{\Gamma(\gamma)}{\Gamma(2\gamma)\Gamma(1/2-\gamma)} 
> +\frac{2\Gamma(1-2\gamma)}{\Gamma(1-\gamma)\Gamma(1/2-\gamma)}}\right)^2 
> \left(\frac{\Gamma(\gamma)\Gamma(1/2 - \gamma/ 2)}{\Gamma(\gamma 
> /2)}\right)^4
>
>which you can see displayed on http://oeis.org/wiki/User:Peter_Luschny
>
>Peter (... sorry, could not resist.)
>
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