1/2+1/(1+2)+1/(1+2+3)+.....+1/(1+...+100)=?
1/2+1/[1+2]+1/[1+2+3]+...+1/[1+2+3..+100]
=1/2+2*[1/2-1/3]+2[1/3-1/4]+...+2[1/100-1/101]
=1/2+2[1/2-1/3+1/3-1/4+...+1/100-1/101]
=1/2+2[1/2-1/101]
=1/2+1-2/101
=299/202
1/2+1/[1+2]+1/[1+2+3]+...+1/[1+2+3..+100]
=1/2+2*[1/2-1/3]+2[1/3-1/4]+...+2[1/100-1/101]
=1/2+2[1/2-1/3+1/3-1/4+...+1/100-1/101]
=1/2+2[1/2-1/101]
=1/2+1-2/101
=299/202