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