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December 3rd, 2009, 02:19 PM  #1 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Pi formula
Well, I've done it. I've doubled the speed of the Leibniz series for pi, while finding a formula for the denominators. 8/3 + 8/35 + 8/99 + ... Let p be the previous denominator. Let t be the number of terms so far. n = p + (32*(t1)) Now, I'm having trouble doubling it AGAIN, and finding formulae for it. Apparently, it still has 32, attached to it. 304/105 + 2,352/19,305 + 6,448/156,009 + ... So, I need some help, here. 
December 5th, 2009, 03:02 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
Your results are correct, but how did you obtain the formula you gave? Why can't you just apply the same method again to obtain a second formula? In your last series, the nth numerator (starting at n = 0) is given by 16(16(2n + 1)² + 3). 
December 10th, 2009, 07:11 AM  #3 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Re: Pi formula
Well, this is how I did it: 8/3 + 8/35 + 8/99 + 8/195 + ... 35 = 3 + 32 99 = 35 + 64 195 = 99 + 96 64 = 32*2 96 = 32*3 35 = 2nd 99 = 3rd 195 = 4th So: p + (32*(n1)) = denominator And that is how I did it. 
December 11th, 2009, 03:42 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
How did you obtain "8/3 + 8/35 + 8/99 + 8/195 + ..." in the first place?

December 15th, 2009, 05:41 PM  #5 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Re: Pi formula
4  4/3 + 4/5  4/7 + 4/9  4/11 + 4/13  4/15 + ... = Pi 4  4/3 = 8/3 4/5  4/7 = 8/35 4/9  4/11 = 8/99 4/13  4/15 = 8/195 Therefore: 8/3 + 8/35 + 8/99 + 8/195 + ... = Pi 
December 17th, 2009, 06:41 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
If you sum the terms in groups of four instead of two, you get the second series. I gave the expression for the numerator, which is obtainable by working out 4/(8n + 1)  4/(8n + 3) + 4/(8n + 5)  4/(8n + 7).

January 21st, 2010, 01:57 PM  #7 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Re: Pi formula
I found the formula for the numerator! It's this: Code: p + (2^(t+9)) 
January 23rd, 2010, 12:48 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
That's incorrect. The correct formula is p + 2048(t  1), which is easily derived from the result I gave earlier.

January 23rd, 2010, 10:18 AM  #9 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Re: Pi formula
Whoops! Works for the first 2 differences.

February 12th, 2010, 10:23 AM  #10 
Newbie Joined: Dec 2009 Posts: 15 Thanks: 0  Re: Pi formula
Okay, this is the finished product. Starting with n = 0: 16(16(2n+1)^2 + 3) ______________________________________ 4096n^4 + 8192n^3 + 5504n^2 + 1408n + 105 Sorry about the double post, it wouldn't let me edit! _ 

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