(1-1/2²)(1-1/3²)(1-1/4²).(1-1/99²)(1-1/100²)
=(2²-1)/2²*(3²-1)/3²*(4²-1)/4²*.(99²-1)/99²*(100²-1)/100²
=(1*3/2²)*(2*4/3²)*(3*5/4²)*.(98*100/99²)*(99*101/100²)
=(1*2*3*4*.98*99)*(3*4*5*6*.100*101)/(2²*3²*4²*.99²*100²)
=(2*3*4*.98*99*100/100)*(2*3*4*5*6*.100*101/2)/(2*3*4*.99*100)²
=(2*3*4*.99*100)²*101/(100*2)/(2*3*4*.99*100)²
=101/200
或
(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/99²)(1-1/100²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4).(1-1/99)(1+199)(1-1/100)(1+1/100)
=1/2*3/2*2/3*4/3*3/4*5/4*.98/99*100/99*99/100*101/100
=1/2*101/100
=101/200 (1-1/2²)(1-1/3²)(1-1/4²).(1-1/99²)(1-1/100²)
=(2²-1)/2²*(3²-1)/3²*(4²-1)/4²*.(99²-1)/99²*(100²-1)/100²
=(1*3/2²)*(2*4/3²)*(3*5/4²)*.(98*100/99²)*(99*101/100²)
=(1*2*3*4*.98*99)*(3*4*5*6*.100*101)/(2²*3²*4²*.99²*100²)
=(2*3*4*.98*99*100/100)*(2*3*4*5*6*.100*101/2)/(2*3*4*.99*100)²
=(2*3*4*.99*100)²*101/(100*2)/(2*3*4*.99*100)²
=101/200
或
(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/99²)(1-1/100²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4).(1-1/99)(1+199)(1-1/100)(1+1/100)
=1/2*3/2*2/3*4/3*3/4*5/4*.98/99*100/99*99/100*101/100
=1/2*101/100
=101/200