Question

求证：1+1/2+1/3+···+1/2^n＞n+2/2 (n≥2）

Answer 1

证明：

(1)当n=2时,

左边=1 + 1/2 + 1/3 + 1/4 = 25/12

右边= (2+2)/2 = 2 = 24/12

所以左边>右边成立,即n=2时命题成立.

(2)假设当n=k (k>=2时)命题成立,

即1+1/2+1/3+...+1/2^k > (k+2)/2

则当n=k+1时,

左边 = 1+1/2+1/3+...+1/2^k + 1/(2^k + 1) + ...+ 1/2^(k+1)

> (k+2)/2 + 1/2^(k+1) + 1/2^(k+1) + ...+ 1/2^(k+1)

= (k+2)/2 + 2^k / 2^(k+1)

= (k+2)/2 + 1/2

= (k+1 +2)/2

即n=k+1时也成立.

由(1)(2)可得原命题成立