证明：∵

1

n(n+1)(n+2)

=

1

2

[（

1

n

-

1

n+1

）-（

1

n+1

-

1

n+2

）]，
∴

1

1×2×3

+

1

2×3×4

+…+

1

n(n+1)(n+2)

=

1

2

[（1-

1

2

）-（

1

2

-

1

3

）]+…+

1

2

[（

1

n

-

1

n+1

）-（

1

n+1

-

1

n+2

）]=
=

1

2

[（1-

1

2

）-（

1

n+1

-

1

n+2

）]＜

1

4

．