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January 8th, 2015, 10:36 AM  #1 
Newbie Joined: Jan 2015 From: Kashmir, India Posts: 2 Thanks: 0  Formulas for Pi I've found!
Hello everyone! First, I'd like to introduce myself. I'm Syed Fahad (but my first name is Fahad), a teen programmer, and an aspiring mathematician. I'm writing a paper on convergence improvement of series using a special method I haven't named yet. I've got some amazingly rapid series for, among others, pi. If these series seem interesting, please tell me, and only then will I continue working on my paper. Else, it's just be a serious waste of time for a 14 year old teen like me. Here are two series I've found: 1. Based on Ramanujan's: 1/pi=sqrt(2)/120509459123328(seriesSum((20766825650421760k^5+41 967637769957376k^4+32018211447629056k^3+1103439257 7777792k^2+1514919861224944k+27124158075000)(8k)!/(((2k+1)!)^4*396^8k), k, 0, infinity)) This should give 16 digits per term. 2. Based on Chudnovsky's: 1/pi=12(seriesSum((494621615140994140579908108288k^7 +1242719979646474354834428936192k^6+12794478709564 01489285132642304k^5+68899238297099074267771862016 0k^4+204180254546238928976370553152k^3+31637100076 081839186422223168k^2+2081398005690255816499272816 k+21409520118014285485961040)(12k)!/((6k+3)!((2k+1)!)^3*262537412640768000^2k+3/2), k, 0, infinity)) And the one that removes the fractional power: 1/pi=sqrt(10005)(seriesSum((494621615140994140579908 108288k^7+1242719979646474354834428936192k^6+12794 47870956401489285132642304k^5+68899238297099074267 7718620160k^4+204180254546238928976370553152k^3+31 637100076081839186422223168k^2+2081398005690255816 499272816k+21409520118014285485961040)*(12k)!/((6k+3)!*68925893036108889235415629824000000^k((2k +1)!)^3), k, 0, infinity))/1121280066934450893619200000 Both of them should give 2829 digits per term. I hope you find them interesting enough to try out. If you face any problems in trying them out, tell me and I'll paste the MathML code for these formulas (that will look better ) I don't know what class no. they have, what discriminant etc... I have many other private series for arctan(x), e, exp(x), sin/cos/tan, and reimann zeta function to name a few. And for pi, the formulas I've found based on Borwein's series give upto 100 digits per term! And I repeat, if these are interesting for the Math world, do tell me so that I'd continue working on that paper. 
January 8th, 2015, 10:56 AM  #2 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 884 Thanks: 61 Math Focus: सामान्य गणित 
great job keep it up 
January 8th, 2015, 11:06 AM  #3 
Newbie Joined: Jan 2015 From: Kashmir, India Posts: 2 Thanks: 0  

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