(1-1/2²)(1-1/3²)(1-1/4²)……(1-1/2010²)

问题描述:

(1-1/2²)(1-1/3²)(1-1/4²)……(1-1/2010²)

={(2+1)(2-1)}/(2×2)×{(3+1)(3-1)}/(3×3)×……×{(2010+1)(2010-1)}/(2010×2010)
=(1×3)/(2×2)×(2×4)/(3×3)×……×(2009×2011)/(2010×2010)
=1/2×2011/2010
=2011/4020