My Math Forum Pi formula

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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*(t-1)) 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: 20,926 Thanks: 2205 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*(n-1)) = denominator And that is how I did it.
 December 11th, 2009, 03:42 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 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: 20,926 Thanks: 2205 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)) Thanks, skipjack, for the other formula!
 January 23rd, 2010, 12:48 AM #8 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 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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