Question

n∈Z⁺，則

1

1×3

+

1

3×5

+

1

5×7

+…+

1

(2n-1)•(2n+1)

=

 

．

Answer 1

考點(diǎn)：數(shù)列的求和

專題：等差數(shù)列與等比數(shù)列

分析：直接利用裂項(xiàng)相消法求數(shù)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)

．
故答案為：

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

4n(2n+1)

．

點(diǎn)評(píng)：本題考查了裂項(xiàng)相消法求數(shù)列的和，是中檔題．