Wallis Formula

Author

Eric W. Weisstein
January 19, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/WallisFormula.nb.

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Products

(∏_(n=1)^∞)(1-((x^2)/((π^2) (n^2))))==sinc(x)

V6.0.3

Product[1-((x^2)/((π^2)(n^2))),{n,∞}]

V7

Product[1-((x^2)/((π^2)(n^2))),{n,∞}]

(∏_(n=1)^∞)1-(1/(((2 n))^2))==(π/2)

NProduct[1-(1/(((2n))^2)),{n,∞}]

V6

Product[1-(1/(((2n))^2)),{n,∞}]

V7.0

Product[1-(1/(((2n))^2)),{n,∞}]

V7.0.1

Product[1-(1/(((2n))^2)),{n,∞}]

(∏_n^∞)((((2 n))^2)/((2 n-1) (2 n+1)))==(π/2)

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

V6

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

V7.0

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

V7.0.1

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

(1/((∏_k^∞)(1-((((1/2)))^k))))==(1/(QPochhammer2[(1/2), (1/2)]))

1/Product[(1-((((1/2)))^k)),{k,∞}]

FullSimplify[%]

NProduct[(((1-(2^(-k))))^(-1)),{k,∞}]

1/NProduct[(1-(((1/2))^k)),{k,∞}]

Sums

(∑_n^∞)(∑_(k=0)^n)(((((-1))^(k+1)) log(k+1) (Binomial[n, k]))/(2^n))==log((π/2))

Sum[((((-1))^(k+1))/(2^n))Binomial[n,k]Log[k+1],{n,20},{k,0,n}]//N

Log[π/2.]

Exp@Table[Sum[(((-1))^(k+1))Binomial[n,k]Log[k+1],{k,0,n}],{n,5}]

N[Exp[Sum[(1/(2^n))(((-1))^(k+1))Binomial[n,k]Log[k+1],{n,100},{k,0,n}]],20]//Timing

N[Pi/2,20]

V6.0.3

Sum[(((-1))^(k+1))Binomial[n,k]Log[k+1],{k,0,n}]//Timing

V7

TimeConstrained[Sum[(((-1))^(k+1))Binomial[n,k]Log[k+1],{k,0,n}],300]//Timing

TimeConstrained[Sum[((((-1))^(k+1))/(2^n))Binomial[n,k]Log[k+1],{n,∞},{k,0,n}],300]//Timing

Sum[((((-1))^(k+1))/(2^n))Binomial[n,k]Log[k+1],{n,10},{k,0,n}]//N

Log[Pi/2.]

(1/2) (∑_n^∞)(((-1))^(n-1)) ((n^(-s))-(((n+1))^(-s)))+(1/2)==(PolyLog[s, -1])

(1/2)+(1/2)Sum[(((-1))^(n-1))((n^(-s))-(((n+1))^(-s))),{n,∞}]

FullSimplify[%]

FullSimplify[%+PolyLog[s,-1]]

FunctionExpand[PolyLog[s,-1]]

(1/2) (∑_n^∞)(((-1))^n) (((log(n))/(n^s))-((log(n+1))/(((n+1))^s)))==(2^(-s)) (log(4) (Zeta[s])+((2^s)-2) ζ'(s))

(1/2)Sum[(((-1))^n)(((Log[n])/(n^s))-((Log[n+1])/(((n+1))^s))),{n,∞}]

FullSimplify[%,s>0]

%/.s->0//FullSimplify

(1/2) (∑_n^∞)(((-1))^n) (log(n)-log(n+1))==(1/2) log((π/2))

(1/2)Sum[(((-1))^n)(Log[n]-Log[n+1]),{n,∞}]

(1/2)NSum[(((-1))^n)(Log[n]-Log[n+1]),{n,∞}]

(1/2)Log[π/2.]

Identities

((((4^(ζ(0))) (e^(-ζ'(0)))))^2)==(π/2)

((4^(Zeta[0]))Exp[-Zeta'[0]])^2

{Zeta[0],Zeta'[0]}

Yung Derivation

F(s)

Simplify[1/2+1/2Sum[(-1)^(n-1)(n^(-s)-(n+1)^(-s)),{n,∞}]]

Collect[Expand[Simplify[1/2+1/2Sum[(-1)^(n-1)(n^(-s)-(n+1)^(-s)),{n,∞}]]],Zeta[s]]

-PolyLog[s,-1]

F'(s)

FullSimplify[D[(-1)^(n-1)(n^(-s)-(n+1)^(-s)),s]]

%/.s->0

Table[(((-1))^n) (Log[n]-Log[1+n]),{n,5}]

Collect[Expand[1/2Sum[(((-1))^(-1+n)) (-(n^(-s)) Log[n]+(((1+n))^(-s)) Log[1+n]),{n,∞}]]/.Log[4]:>2Log[2],Zeta'[s]]

D[(1-(2^(1-s))) Zeta[s],s]