f(x)=1/(x^3+3x^2+2x),f(1)+f(2)+f(3)+…+f(n)>502/2014,求满足不等式成立的最小n值.

题目

f(x)=1/(x^3+3x^2+2x),f(1)+f(2)+f(3)+…+f(n)>502/2014,求满足不等式成立的最小n值.

答案

f(x)=1/x(x+1)(x+2) = 1/x(x+1) - 1/x(x+2) = 1/x-1/(x+1)-1/2*(1/x-1/(x+2)) =
f(1) = 1/1-1/2-1/2(1/1-1/3)
f(2)=1/2-1/3-1/2(1/2-1/4)
f(3)=1/3-1/4-1/2(1/3-1/5)
f(n)=...
f(1)+f(2)+f(3)+...+f(n) = 1/1-1/(n+1)-1/2*(1+1/2-1/(n+1)-1/(n+2))
=1/4-1/(n+1)+1/2*(1/(n+1)+1/(n+2)) >502/2014
=>1/(n+2)-1/(n+1)>1004/2014-1/2 = -3/2014
(n+1)-(n+2)>-3(n+1)(n+2)/2014
=>2014<(n+1)(n+2)
=>n >= 44
最小是44