1 +1/(1+2) +1/(1+2+3) +1/(1+2+3+4) +...+1/(1+2+3+...+100)
=1 +1/[(1+2)*2/2] +1/[(1+3)*3/2] +1/[(1+4)*4/2] +...+1/[(1+100)*100/2]
=1 +2/(3*2) +2/(4*3) +2/(5*4) +...+2/(101*100)
=1 +2*[1/(3*2) +1/(4*3) +1/(5*4) +...+1/(101*100)]
=1 +2*[(1/2 -1/3) +(1/3 -1/4) +(1/4 -1/5) +...+(1/100 -1/101)]
=1 +2*(1/2 -1/101)
=1 +1 -2/101
=200/101
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