Question

计算：（1-1/2²）（1-1/3²）... ...（1-1/2005²）（1-1/2006²）

Answer 1

解:(1-1/2²)(1-1/3²)(1-1/4²)…(1-1/2005²)(1-1/2006²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)(1-1/5)(1+1/5)(1-1/6)(1+1/6)(1-1/7)(1+1/7)(1-1/8)(1+1/8)……(1-1/2005)(1+1/2005)(1-1/2006)(1+1/2006)
=1/2×3/2×2/3×4/3×3/4......2004/2005×2006/2005×2005/2006×2007/2006
-----注意：除了第1分数和最后一个分数外，全约分了。
=1/2×2007/2006
=2007/4012

Answer 2

解:(1-1/2²)(1-1/3²)(1-1/4²)…(1-1/2005²)(1-1/2006²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)(1-1/5)(1+1/5)(1-1/6)(1+1/6)(1-1/7)(1+1/7)(1-1/8)(1+1/8)……(1-1/2005)(1+1/2005)(1-1/2006)(1+1/2006)
=(1/2*2/3*3/4...2005/2006)(3/2*4/3*5/4...2007/2006)
=1/2006*2007/2
=2007/4012

Answer 3

解:(1-1/2²)(1-1/3²)(1-1/4²)…(1-1/2005²)(1-1/2006²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)(1-1/5）……(1-1/2005)(1+1/2005)(1-1/2006)(1+1/2006)
=1/2*3/2*2/3*4/3*3/4*5/4*4/5...2004/2005*2006/2005*2005/2006*2007/2006
=1/2*2007/2006
=2007/4012