Question

(1-1/2^2)(1-1/3^2)...(1-1/2005^2)

Answer 1

(1-1/2^2)*(1-1/3^2)*(1-1/4^2)......(1-1/2005^2)^2
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)...(1-1/2005)(1+1/2005)
=1/2*3/2*2/3*4/3*3/4*5/4*...*2004/2005*2006/2005
=1/2*2006/2005
=1003/2005
或
通式为(1-1/2^2)(1-1/3^2）（1-1/4^2)……（1-1/（n-1)^2)(1-1/n^2)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)……（1+1/（n-1））（1-1/（n-1））（1+1/n）（1-1/n）
={（n+1）/2}（1/n）=（n+1）/2n
n=2005
所以结果=1003/2005

Answer 2

(1-1/2^2)(1-1/3^2)...(1-1/2005^2)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)...(1-1/2005)(1+1/2005)
=(1/2)(3/2)(2/3)(4/3)...(2004/2005)(2006/2005)
=1/2*2006/2005
=1003/2005

Answer 3

(1-1/2^2)(1-1/3^2)...(1-1/2005^2)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)...(1-1/2005)(1+1/2005)
=(1/2)(3/2)(2/3)(4/3)...(2004/2005)(2006/2005)
=1/2*2006/2005
=1003/2005