1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2001)

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1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2001)1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2

1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2001)
1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2001)

1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2001)
原式=2/(2*3)+2/(3*4)+2/(4*5)+.+2/(2001*2002)
=2*[1/2-1/3+1/3-1/4+1/4-1/5+..+1/2001-1/2002]
=2*(1/2-1/2002)
=1-1/1001
=1000/1001