Question

1道数列极限题

计算 lim n^2(k/n-1/(n+1)-1/(n+2)-......-1/(n+k))
1楼回答不正确 无数个极限为0加起来不等于0

Answer 1

lim n^2(k/n-1/(n+1)-1/(n+2)-......-1/(n+k))
=lim n^2{[1/n-1/(n+1)]+[1/n-1/(n+2)]......+[1/n-1/(n+k)]
=lim n^2/[n(n+1)]+n^2*2/[n(n+2)]+......+n^2*k/[n(n+k)]
=1+2+......k
=k*(k+1)/2

Answer 2

分子分母同除以n
lim n^2(k/n-(1/n)/(1+1/n)-(1/n)/(1+2/n)-......-(1/n)/(1+k/n))
1/n的极限是0
即lim n^2(0+0+0...+0)=lim0=0