Question

极限题，如图

不清楚答案是多少，但不是0

Answer 1

原式=∑[i=1:n]1/n (i/n)^5
=∫x^5 dx
=1/6 x^6|[0,1]
=1/6 (1-0)
=1/6
转化成定积分的定义公式

Answer 2

an
=n^5
=(n-2)(n-1)n(n+1)(n+2) + 5n^3 -4n
let

bn = (n-2)(n-1)n(n+1)(n+2) , cn =5n^3 , dn =-4n
bn
=(n-2)(n-1)n(n+1)(n+2)
=(1/6) [(n-2)(n-1)n(n+1)(n+2)(n+3)-(n-3)(n-2)(n-1)n(n+1)(n+2) ]
Bn
=b1+b2+...+bn
=(1/6)(n-2)(n-1)n(n+1)(n+2)(n+3)
cn
=5n^3
=5(n-1)n(n+1) +5n
=(5/4) [(n-1)n(n+1)(n+2)-(n-2)(n-1)n(n+1)] + (5/2)[ n(n+1) -(n-1)n]
Cn
=c1+c2+...+cn
=(5/4)(n-1)n(n+1)(n+2) + (5/2)n(n+1)
=(5/4)n(n+1).[ (n-1)(n+2) + 2 ]
=(5/4)n(n+1).[n(n+1)]
=(5/4)[n(n+1)]^2
dn=-4n
Dn
=d1+d2+...+dn
=-2n(n+1)
an =bn +cn +dn
Sn
=a1+a2+...+an
=Bn +Cn + Dn
=(1/6)(n-2)(n-1)n(n+1)(n+2)(n+3) +(5/4)[n(n+1)]^2 -2n(n+1)
coef. of n^6 of Sn =1/6
lim(n->∞) ( 1^5 +2^5+....+n^5) /n^6
=lim(n->∞) (1/6)n^6 /n^6
=1/6