Question

求n趋于无穷大时[1*2+2*3+...+n*(n+1)]/n的三次方的极限

Answer 1

1*2+2*3+...+n*(n+1)

=1^+1+2^+2+…+n^+n

=1+2+…n+1^+2^+…+n^

=n(n+1)/2+n(n+1)(2n+1)/6

=n^3/3+n^+2n/3

lim{[1*2+2*3+...+n*(n+1)]/n^3}

=lim[(n^3/3+n^+2n/3)/n^3]

=1/3