Question

[1/（n的平方十n+1）〕十……+n/（n的平方十n十n）、n趋于无穷,极限

Answer 1

n^2 为n的平方,inf为正无穷 J(a,b) f(x) dx 表示 f(x)从a到b的积分

设 f(k) = k/(n^2 +n + k) ,g(n) = f(1)+ f(2) +...+f(n)则

J(k,k+1) f(x) dx < f(k) 0

J(1,n + 1) f(x)dx =n - (n^2+n)ln(1+n/(n^2+n+1))

= n -(n^2+n) (n / (n^2+n+1)-(n / (n^2+n+1))^2 / 2+O((n / (n^2+n+1))^3))

= n - n + 1/2 -O(1/n) =1/2,n->inf

同理J(0,n)f(x)dx = n -(n^2+n)ln(1+ n/(n^2+n)) = n - n +1/2 -O(1/n)= 1/2,n->inf

所以

J(1,n+1) f(x) dxinf,即

1/(n^2+n+1)+ ...+n/(n^2+n+n) = 1/2,n->inf