数列证明!若N为大于1的自然数,证明：1/N+1+1/N+2+...+1/2N>13/24

Question

数列证明!
若N为大于1的自然数,证明：1/N+1 +1/N+2 +...+1/2N >13/24

Answer 1

证明：数学归纳法
Sn=1/N+1 +1/N+2 +...+1/2N
N=2时,S1=1/3+1/4=7/12>13/24
设N=k时上式成立,即Sk>13/24
则当N=k+1时：
Sk+1=1/k+2 +1/k+3 +...+1/2k+2=1/(k+1)+1/(k+2)+.1/(2k+1)+1/(2k+2)
+1/(2k+2)-1/(k+1)=Sk+1/(2k+1)+1/(2k+2)-1/(k+1)
=Sk+1/(2k+1)-1/(2k+2)>Sk>13/24 (后一项＞0)
所以当N=k+1时也成立.
所以：1/N+1 +1/N+2 +...+1/2N >13/24