1/1+1/（1+2）+1/（1+2+3）+.+1/（1+2+3+.+100）

因为1+2+3+.+n=n(n+1)/2
并且1/n(n+1)
=[(n+1)-n]/n(n+1)
=(n+1)/n(n+1)-n/n(n+1)
=1/n-1/(n+1)
所以,原式=2/(1×2)+2/(2×3)+2/(3×4)+.+2/(100×101)
=2×[1/(1×2)+1/(2×3)+1/(3×4)+.+1/(100×101)]
=2×(1/1-1/2+1/2-1/3+1/3-1/4+.+1/99-1/100+1/100-1/101)
=2×(1-1/101)
=2×100/101
=200/101

1+2+3+……+n = (n+1)n/2
1/(1+2+3+……+n)=2/(n(n+1))=2(1/n-1/(1+n))
原式= 2(1/1-1/2+1/2-1/3+1/3-1/4+-……+1/100-1/101)
= 2(1-1/101)
=200/101