1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)

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1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)

1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)
1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)

1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)
1/x(x+2)+1/(x+2)*(x+4)+1/(x+4)*(x+6)+.+1/(x+2000)*(x+2002)
=1/2(1/x-1/(x+2))+1/2(1/(x+2)-1/(x+4))+1/2(1/(x+4)-1/(x+6))+.+1/2(1/(x+2000)-1/(x+2002))
=1/2(1/x-1/(x+2002))
=1/2(2002/x(x+2002))
=1001/x(x+2002)

=1/2[1/x-1/(x+2)]+1/2[1/(x+2)-1/(x+4)]+[1/(x+4)-1/(x+6)]+......+1/2[1/(x+2000)-1/(x+2002)]
=1/2[1/x-1/(x+2)+1/(x+2)-1/(x+4)+1/(x+4)-1/(x+6)+......+1/(x+2000)-1/(x+2002)]
=1/2[1/x-1/(x+2002)]
=1001/x(x+2002)

=1/2[1/x-1/(x+2)]+1/2[1/(x+2)-1/(x+4)]+[1/(x+4)-1/(x+6)]+......+1/2[1/(x+2000)-1/(x+2002)]
=1/2[1/x-1/(x+2)+1/(x+2)-1/(x+4)+1/(x+4)-1/(x+6)+......+1/(x+2000)-1/(x+2002)]
=1/2[1/x-1/(x+2002)]
=1001/x(x+2002) 应该对的

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