-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)

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-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)-(-1/1x2-1/2x3-1/3

-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)
-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)

-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)
-(-1/1x2-1/2x3-1/3x4-1/4x5-.-1/2001x2002)
=1-1/2+1/2-1/3+.+1/2001-1/2002
=1-1/2002
=2001/2002

结果为2001/2002
这个是有规律的,每项都为-1/(n(n+1)),n为项数,将其拆开为1/(n+1)-1/n,代入原式,每项分为两个式子,此时前一项的第一个式子与后一项的第二个式子抵消一直做下去,最后为:
-(-1+1/2002)=2001/2002