1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)……+1/(1+2+3……+99)+1/(1+2+3……+99+100)=?

1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)……+1/(1+2+3……+99)+1/(1+2+3……+99+100)=?
1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)……+1/(1+2+3……+99)+1/(1+2+3……+99+100)=?

1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)……+1/(1+2+3……+99)+1/(1+2+3……+99+100)=?
1/(1+2+...+n)=1/(n*(n+1)/2)=2/(n*(n+1))=2(1/n-1/(n+1))
1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)……+1/(1+2+3……+99)+1/(1+2+3……+99+100)
=2*(1/1-1/2+1/2-1/3+...+1/100-1/101)
=2*(1-1/101)
=200/101

因为1+2+3+……+99+100=[100(1+100)]/2
所以1/(1+2+3……+99+100)=2/[100(1+100)]=2(1/100-1/101)
同理可知原式中的每个分数都可按上述步骤化为两个分数差的2倍
所以原式=1+2(1/2-1/3+1/3-1/4+1/4-……+1/100-1/101)
=1+2(1/2-1/101)=200/101