Question

n∈Z⁺，则

1

1×3

+

1

3×5

+

1

5×7

+…+

1

(2n-1)•(2n+1)

=

 

．

Answer 1

考点：数列的求和

专题：等差数列与等比数列

分析：直接利用裂项相消法求数列的和．

解答： 解：∵

1

(2n-1)(2n+1)

=

1

2

(

1

2n-1

-

1

2n+1

)，
∴

1

1×3

+

1

3×5

+

1

5×7

+…+

1

(2n-1)•(2n+1)

=

1

2

[(1-

1

3

)+(

1

2

-

1

4

)+(

1

3

-

1

5

)+…+(

1

2n-1

-

1

2n+1

)]
=

1

2

(1+

1

2

-

1

2n

-

1

2n+1

)=

(2n-1)(3n+1)

4n(2n+1)

．
故答案为：

(2n-1)(3n+1)

4n(2n+1)

．

点评：本题考查了裂项相消法求数列的和，是中档题．