Question

高数求极限题目

lim(1/(n^2+n+1)+2/(n^2+n+2)+......+n/(n^2+n+n)),n趋向于正无穷

Answer 1

设 f(n) = 1/(n^2+n+1)+2/(n^2+n+2)+......+n/(n^2+n+n)),
f(n) >= (1+2+3+......+n) / (n^2+n+n) = n*(n+1) /2 / (n^2+n+n) = u(n)
f(n) <= (1+2+3+......+n) / (n^2+n+1) = n*(n+1) /2 / (n^2+n+1) = v(n)
Limit [ u(n), n ->∞] = Limit [ v(n), n ->∞] = 1/2
根据夹挤准则， Limit [ f(n), n ->∞] = 1/2 为所求。

Answer 2

为了方便叙述，原式我就简写为limf(n)了
显然f(n)>(1+2+……n)/(n^2+n+n)=(n^2+n)/(2*(n^2+n+n)………………A
又f(n)<(1+2+……n)/(n^2+n+1)=(n^2+n)/(2*(n^2+n+1)………………B
现在分别求A，B两式极限就很好求了
A，B两式求法相同，均对分子分母除n^2，易得limA=1/2，limB=1/2
即1/2<limf(n)<1/2
故limf(n)=1/2