急!求正整数的最大值,使不等式(1/n+1)+(1/n+2)+...+(1/3n+1）＞a-7,对一切正整数n都成立.

设f(n) = 1/(n+1) + 1/(n+2) + 。。。 + 1/(3n+1)则f(n+1) = 1/(n+2) + 1/(n+3) + 。。。 + 1/[3(n+1)+1] = 1/(n+2) + 1/(n+3) + 。。。 + 1/(3n+4)则f(n)-f(n+1) = 1/(n+1) - [1/(3n+2) + 1/(3n+3) + 1/(3n+4)] = 1/(n+1) - [1/(3n+3)+(3n+2+3n+4)/((3n+2)(3n+4))] = 1/(n+1) - [1/(3n+3) + (6n+6)/((3n+2)(3n+4))]【(3n+2)(3n+4)=9n^2+18n+81 /((3n+3)(3n+3))……应用到下式】f(n)-f(n+1)< 1/(n+1) - [1/(3n+3) + (6n+6)/((3n+3)(3n+3))] = 1/(n+1) - [1/(3n+3) + 2/(3n+3)] = 1/(n+1) - 3/(3n+3) = 0因为f(n)-f(n+1)a-7对所有自然数n成立所以只要13/12>a-7,解得a