(1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)

来源：百度知道 编辑：UC知道 时间：2024/05/11 01:10:46

由n^2-1=(n-1)(n+1)
(1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)
=(2^2-1)/2^2*(3^2-1)/3^2........(100^2-1)/100^2
=(1*3/2^2)*(2*4/3^2)*(3^5/4^2)....99*101/100^2
=1/2*101/100(中间约了很多项)
=101/200

(1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)......(1+1/100)(1-1/100)
=(3/2*4/3*5/4...*101/100)*(1/2*2/3*3/4...99/100)
=101/2*1/100
=101/200.

根据平方差公式，
(1-1/2^2)*(1-1/3^2)*(1-1/4^2).......(1-1/100^2)
=[(3/2)(4/3)......(101/100)]*[(1/2)(2/3)......(99/100)]
=(101/2)*(1/100)
=0.505