1/1×2+1/2×3+1/3×4+...+1/1999×2000=?

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1/1×2+1/2×3+1/3×4+...+1/1999×2000=?1/1×2+1/2×3+1/3×4+...+1/1999×2000=?1/1×2+1/2×3+1/3×4+...+1/1999×2

1/1×2+1/2×3+1/3×4+...+1/1999×2000=?
1/1×2+1/2×3+1/3×4+...+1/1999×2000=?

1/1×2+1/2×3+1/3×4+...+1/1999×2000=?
1/1×2+1/2×3+1/3×4+...+1/1998×1999+1/1999×2000
=1-1/2+1/2-1/3+1/3-1/4+------+1/1998-1/1999+1/1999-1/2000
=1-1/2000
=1999/2000

因为1/n(n+1)=1/n-1/(n+1)，
所以1/1×2+1/2×3+1/3×4+...+1/1999×2000=1-1/2+1/2-1/3+1/3+…-1/1999+1/1999-1/2000
=1-1/2000=1999/2000