Question

用数学归纳法证明:1+1/2+1/3+……+1/2^n>(n+2)/2 (n>=2,正整数)

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.