1/1×2+1/2×3+1/3×4+…+1/2010×2011+1/2012×2013

问题描述:

1/1×2+1/2×3+1/3×4+…+1/2010×2011+1/2012×2013

每一项的分子是n*(n+1)的形式,则1/n*(n+1)=1/n-1/(n+1)
则原式=1-1/2+1/2-1/3+1/3-1/4+1/4+……-1/2012+1/2012-1/2013=1-1/2013=2012/2013