Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Prime numbers and pi
Replies: 33   Last Post: Dec 10, 2012 3:43 PM

 Messages: [ Previous | Next ]
 Franz Gnaedinger Posts: 330 Registered: 4/30/07
Re: Prime numbers and pi
Posted: Apr 18, 2009 2:22 PM

'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.