计算(1-1/2^2)(1-1/3^2)(1-1/4^2)…(1-1/2009^2)的值

题目

计算(1-1/2^2)(1-1/3^2)(1-1/4^2)…(1-1/2009^2)的值

答案

(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/2009²)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)...(1+1/2009)(1-1/2009)
=(3/2)(1/2)(4/3)(2/3)(5/4)(3/4)...(2010/2009)(2008/2009)
=[(1/2)(2/3)(3/4)...(2008/2009)][(3/2)(4/3)(5/4)...(2010/2009)]
=[(1×2×3×...×2008)/(2×3×4×...×2009)]×[(3×4×5×...×2010)/(2×3×4×...×2009)]
=(1/2009)×(2010/2)
=1005/2009