Pippenger Product

Author

Eric W. Weisstein
January 24, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/PippengerProduct.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Formula

(∏_n^∞)((((((((((2^(n-1))-1)!!))^2) ((((2^n)!!))^2))/(2 ((((2^(n-1))!!))^2) (((((2^n)-1)!!))^2)))))^(1/(2^n)))==(e/2)

Table[(((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2))),{n,5}]

Times@@Table[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,10}]//N

E/2.

TimeConstrained[NProduct[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,∞}],300]//Timing

TimeConstrained[Product[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,∞}],300]//Timing

2 (Sqrt[2]) (∏_(n=2)^∞)((((((2^((2^n)-1)) ((Gamma[(1/2)+(2^(n-2))])^4))/(π ((Gamma[(1/2)+(2^(n-1))])^2)))))^(1/(2^n)))==(e/2)

2N[((((2/1)))^(1/2))((((2/3)(4/3)))^(1/4))((((4/5)(6/5)(6/7)(8/7)))^(1/8))]

FullSimplify[((((((2^(n-1))-1)!!)^2)((((2^n))!!)^2))/(2 (((2^(n-1))!!)^2)((((2^n)-1)!!)^2))),n∈Integers&&n>2]

V6

2Sqrt[2]NProduct[((((((2^((2^n)-1))((Gamma[1/2+(2^(n-2))])^4))/(π ((Gamma[1/2+(2^(n-1))])^2)))))^(1/(2^n))),{n,2,15},WorkingPrecision->20]

N[E,20]

2Sqrt[2]Product[((((((2^((2^n)-1))((Gamma[1/2+(2^(n-2))])^4))/(π ((Gamma[1/2+(2^(n-1))])^2)))))^(1/(2^n))),{n,2,∞}]

...wrong...

2Sqrt[2]NProduct[((((((2^((2^n)-1))((Gamma[1/2+(2^(n-2))])^4))/(π ((Gamma[1/2+(2^(n-1))])^2)))))^(1/(2^n))),{n,2,∞}]

...wrong...

V7

2Sqrt[2]Product[((((((2^((2^n)-1))((Gamma[1/2+(2^(n-2))])^4))/(π ((Gamma[1/2+(2^(n-1))])^2)))))^(1/(2^n))),{n,2,∞}]//Timing

TimeConstrained[2Sqrt[2]NProduct[((((((2^((2^n)-1))((Gamma[1/2+(2^(n-2))])^4))/(π ((Gamma[1/2+(2^(n-1))])^2)))))^(1/(2^n))),{n,2,∞}],60]//Timing