(1-1/2²)(1-1/3²)(1-1/4²)........(1-1/100²)

(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+1/99)(1-1/100)(1+1/100）

=3/2*1/2*4/3*2/3...100/99*99/100*101/100

=（1/2）*（101/100）

=101/200
你可以参考下
摘自：http://zhidao.baidu.com/question/24233849.html

(1-1/2²)(1-1/3²).....(1-1/100^2)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)....(1-1/100)(1+1/100)
=(1/2)*(3/2)*(2/3)*(4/3)..*(99/100)*(101/100)
=101/200

用平方差
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)……(1-1/100)(1+1/100)
=(1/2)(3/2)(2/3)(4/3)……(99/100)(101/100)
中间约分约去
=(1/2)(101/100)
=101/200