BBP Formula

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/BBPFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BBPFormula.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

π

π=(∑_(n=0)^∞)(-(2/(8 n+4))-(1/(8 n+5))-(1/(8 n+6))+(4/(8 n+1)))((((1/16)))^n)

Sum[(4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6))/16^n,{n,0,Infinity}]

FullSimplify[%]

π=(∑_(k=0)^∞)((r/(8 k+7))-((8 r)/(8 k+2))-((4 r)/(8 k+3))+((8 r+4)/(8 k+1))-((8 r+2)/(8 k+4))-((2 r+1)/(8 k+5))-((2 r+1)/(8 k+6)))((((1/16)))^k)

(∑_(k=0)^∞)(((r/(8 k+7))-((8 r)/(8 k+2))-((4 r)/(8 k+3))+((8 r+4)/(8 k+1))-((8 r+2)/(8 k+4))-((2 r+1)/(8 k+5))-((2 r+1)/(8 k+6)))/(16^k))

Sum[((4+8r)/(8k+1)-8r/(8k+2)-4r/(8k+3)-(2+8r)/(8k+4)-(1+2r)/(8k+5)-(1+2r)/(8k+6)+r/(8k+7))/16^k,{k,0,Infinity}]

FullSimplify[%]

(π^2)

(Sqrt[(1/8) (∑_(k=0)^∞)((-(216/(((6 k+2))^2))-(72/(((6 k+3))^2))-(54/(((6 k+4))^2))+(9/(((6 k+5))^2))+(144/(((6 k+1))^2)))/(64^k))])

Simplify[Sqrt[Sum[(144/(6k+1)^2-216/(6k+2)^2-72/(6k+3)^2-54/(6k+4)^2+9/(6k+5)^2)/64^k,{k,0,Infinity}]/8]]

N[%,20]

√((∑_(k=0)^∞)(1/(16^k))(-(16/(((8 k+2))^2))-(8/(((8 k+3))^2))-(16/(((8 k+4))^2))-(4/(((8 k+5))^2))-(4/(((8 k+6))^2))+(2/(((8 k+7))^2))+(16/(((8 k+1))^2))))

Simplify[Sqrt[Sum[(16/(8k+1)^2-16/(8k+2)^2-8/(8k+3)^2-16/(8k+4)^2-4/(8k+5)^2-4/(8k+6)^2+2/(8k+7)^2)/16^k,{k,0,Infinity}]]]