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Re: Prime numbers and pi
Posted:
Apr 18, 2009 2:22 PM
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'1 '4 '9 '16 '25 '49 '64 '81 '100 ... = pi x pi / 6
= 4/3 x 9/8 x 25/24 x 49/48 x 121/120 X 169/168 ...
The infinite series can be given as a stairway:
"1
'1 "3
'1 '4 "5
'1 '4 '9 "7
'1 '4 '9 '16 "9
'1 '4 '9 '16 '25 "11
Now multiply the rows of this stairway with the ever
longer partial product in the inverse form:
"1 times 3/4 = 1.5
'1 "3 times 3/4 x 8/9 = 1.111...
'1 '4 "5 times 3/4 x 8/9 x 24/25 = 1.056
'1 '4 '9 "7 times 3/4 ... 48/49 = 1.03245...
'1 '4 '9 '16 "9 times 3/4 ... 120/121 = 1.02330...
'1 '4 '9 ... "11 times 3/4 ... 168/169 = 1.01700...
'1 '4 '9 ... "13 times 3/4 ... 288/289 = 1.01336...
'1 '4 '9 ... "15 times 3/4 ... 260/361 = 1.01049...
Using this method you can easily check the identity
of the infinite series and the infinite product,
first example of Euler's zeta function.
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