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Re: Prime numbers and pi
Posted:
Apr 16, 2009 1:14 AM


Euler's infinite product again:
pi = 2 x 3/2 x 5/6 x 7/6 x 11/10 x 13/14 x 17/18 x ...
The analogous infinite product using all odd numbers in the numerator seems to approximate the natural logarithm of 2:
ln2 = 1/2 x 3/2 x 5/6 x 7/6 x 9/10 x 11/10 ...
Multiplying pairs of factors:
ln2 = 3/4 x 35/36 x 99/100 x 143/144 x 195/196 ...
This product can be given as a stairway approximation:
3/4 x 14/15 3/4 x 35/36 x 19/20 3/4 x 35/36 x 99/100 x 24/25 3/4 x 35/16 x 99/100 x 143/144 x 29/30 3/4 x 35/36 x 99/100 x 143/144 x 195/196 x 34/35
Multiplying the factors of Euler's infinite product:
pi/4 = 3/4 x 35/36 x 143/140 x 323/324 x 667/660 ...
In this form we can give a stairway approximation:
3/4 x 9/8 3/4 x 35/36 x 14/13 3/4 x 35/36 x 143/140 x 19/18 3/4 x 35/36 x 143/140 x 323/324 x 24/23 3/4 x 35/36 x 143/140 x 323/324 x 667/660 x 29/28
The numbers of the last terms increase by five in both stairways, but while the rows of the first stairway approximate ln2 smoothly from above, the rows of the second stairway oscillate around one fourth of pi.



