高考数学_压轴题_放缩法技巧全总结

2013高考数学备考

一、裂项放缩

nn

例1.(1)求 2的值; (2)求证: 1 5.

22

3k 14k 1k 1k解析:(1)因为 (2)因为

24n 1

1

n2

2

211,所以n212n 1 2

(2n 1)(2n 1)2n 12n 12n 12n 1k 14k 1

4

n

1111 25 1 ,所以 1 1 2 1 1 2 2 2 2n 12n 1 33 35k 1k4n 1 2n 12n 1 n2

4

1

奇巧积累:(1)

1441 1 2 2 2 2n4n4n 1 2n 12n 1

r 1

r

Cn

(2)

1211

2

CCn(n 1)n(n 1)n(n 1)n(n 1)

1n 1

(3)T

1n!11111 r (r 2) r

r!(n r)!nr!r(r 1)r 1rn

(4)(1 1)n 1 1 1 1

n

2 1

3 215

n(n 1)2

1

n 2 n n 2

2n 1

2n 3 2

11

n 1

(2n 1) 2(2n 3) 2n

(5)

111

n n

n

2(2 1)2 12

n

(6)

21 1 (7)2(n 1 ) 1 2(n n 1) (8) n

n

(9)

111 111 11 ,

k(n 1 k) n 1 kk n 1n(n 1 k)k 1 nn 1 k

n11

(n 1)!n!(n 1)!

(10) (11)

1n

2(2n 1 2n 1)

222n 1 n 1

n

211 n 22

(11) (12) (13) (14)

2n2n2n2n 111 n n n n 1 n(n 2)n2nnn 1

(2 1)(2 1)(2 1)(2 1)(2 2)(2 1)(2 1)2 12 11n

3

1n n

2

111

n(n 1)(n 1) n(n 1) n(n 1) 1 n 1

1

1 n 1 n 1 1

n 1 2n n 1

1n 1

1 1

2n12n

n 32 13

2n 1 2 2n (3 1) 2n 3 3(2n 1) 2n 2n 1 k 211

k! (k 1)! (k 2)!(k 1)!(k 2)!

(15)

1

n n 1(n 2) n(n 1)

22

(15) i 1 j 1

i2 j2

(i j)( 1

2

i j

j 1)

2

i ji 1

2

j 1

2

1

例2.(1)求证:1 1 1

22

3

5

171

(n 2) 2

62(2n 1)(2n 1)