1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
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![1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998](/uploads/image/z/574940-20-0.jpg?t=1%2F+3%2A1%2B+1%2F+4%2A2+%2B+1%2F+5%2A3%2B%E2%80%A6%E2%80%A6%2B1%2F1999%2A1997+%2B+1%2F+2000%2A1998)
1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
=1/2*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6...+1/1998-1/2000)
=1/2*(1+1/2-1/1999-1/2000)
=5993001/7996000
1/ 3*1+ 1/ 4*2 + 1/ 5*3+……+1/1999*1997 + 1/ 2000*1998
=1/2(1-1/3+1/2-1/4+1/3-1/5+...+1/1997-1/1999+1/1998-1/2000)
=1/2(1-1/1999+1/2-1/2000)
=1/2(1998/1999+999/2000)
=999/1999+999/4000
=1/2[(1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+...+(1/1997-1/1999)+(1/1998-1/2000)]
=1/2(1+1/2-1/1999-1/2000)