Question

计算

lim

n→∞

[

1

1×3

+

1

2×4

+

1

3×5

+…+

1

n(n+2)

]=______．

Answer 1

∵2[

1

1×3

+

1

2×4

+…+

1

n(n+2)

]
=1-

1

3

+

1

2

-

1

4

+…+

1

n

-

1

n+2

=1+

1

2

-

1

n+1

-

1

n+2

=

3

2

-

2n+3

(n+1)(n+2)

∴

1

1×3

+

1

2×4

+…+

1

n(n+2)

=

3

4

-

2n+3

2(n+1)(n+2)

∴

lim

n→∞

[

1

1×3

 +

1

2×4

+…+

1

n(n+2)

]=

lim

n→∞

[

3

4

-

2n+3

2(n+1)(n+2)

]=

3

4

故答案为：

3

4