April 15, 2010

Product[a[i],{i,1,10}]

Product[1+z^(2^i),{i,0,2}]  //Expand

Sum[z^j,{j,0,2^3-1}]

Product[1+z^(2^i),{i,0,5}]  //Expand

Sum[ z^j,{j,0,2^6-1}]

Product[1+z^(2^i),{i,0,Infinity}]

Sum[z^j,{j,0,Infinity}]

Product[(i^3-1)/(i^3+1),{i,2,Infinity}]

Product[(k+1)^2/(k(k+2)),{k,1,Infinity}]

Product[1-1/i^2,{i,2,Infinity}]

Product[1-4/i^2,{i,3,Infinity}]

Product[1-x^2/i^2,{i,1,Infinity}]

Product[ 1+x^2/i^2,{i,1,Infinity}]

Product[1-x^2/(i-1/2)^2,{i,1,Infinity}]

Product[1+x^2/(i-1/2)^2,{i,1,Infinity}]

Product[Cos[x/2^i],{i,1,Infinity}]

Table[ {x, Sin[x]/x, Sinc[x]}, {x,.05,.5,.05}]

Sinc[0]

f[n_] :=N[Sin[1]/1 - Product[ Cos[1/2^i],{i,1,n}],20]

f[5]

f[25]

N[ 2/Product[1-1/(4i^2),{i,1,100000}],10]

(Slow Convergence)

?Series

Series[ ArcTan[z], {z,0,50}]

Clear[f]

f[z_] = Normal[ Series[ ArcTan[z],{z,0,50}] ]

ArcTan[1/Sqrt[3]]

N[6*f[1/Sqrt[3]],15]

N[Pi,15]