已知n∈N+,n≥2,求证：cos1/2·cos1/3…cos1/n＞2/3

1.
只需证明ln[cos1/2·cos1/3…cos1/n]>ln[2/3]=-0.405465108..
用常见不等式(大学):
ⅰ.
当0≤x,cosx≥1-x^2/2
ⅱ.
当0≤x≤1/8,ln(1-x)≥-8x/7
还用常见等式:
(1/1^2)+(1/1^2)+(1/2^2)+(1/3^2)+..+(1/n^2)+..=π^2/6
2.
根据1.的不等式和等式得:
当2≤k,ln[cos(1/k)]≥ln[1-1/(2k^2)]≥-(4/7)(1/k^2)
==>
ln[cos1/2·cos1/3…cos1/n]=
=ln[cos(1/2)]+ln[cos(1/3)]+..+ln[cos(1/n)]≥
≥-(4/7)[(1/2^2)+(1/3^2)+..+(1/n^2)]≥
≥-(4/7)[-1+(1/1^2)+(1/2^2)+(1/3^2)+..+(1/n^2)+..]=
=≥-(4/7)[-1+π^2/6]=-0.36853..>
>ln[2/3]=-0.405465108...
==>
cos1/2·cos1/3…cos1/n＞2/3

cos²(1/n)=1-sin²(1/n)>1-1/n²=(n-1)(n+1)/n²
(cos1/2·cos1/3…cos1/n)²>(1·3/2²)(2·4/3²)(3·5/4²)…((n-1)(n+1)/n²)=(n+1)/2n>1/2>4/9
所以cos1/2·cos1/3…cos1/n＞2/3