1/1×2+1/2×3+1/3×4+···+1/2004×2005=

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1/1×2+1/2×3+1/3×4+···+1/2004×2005=
1/1×2+1/2×3+1/3×4+···+1/2004×2005=

1/1×2+1/2×3+1/3×4+···+1/2004×2005=
1/1×2+1/2×3+1/3×4+···+1/2004×2005
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……+(1/2004-1/2005)
=1-1/2+1/2-1/3+1/3-1/4+……+1/2004-1/2005
=1-1/2005
=2004/2005

1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+......1/2003-1/2004+1/2004-1/2005=1-1/2005=2004/2005