Question

lim(n趋近于无穷大）1/n∑(i=1到n)(1+i/n)∧(1/3) 用定积分做。。

-_-!可不可以把第一步写清楚点

Answer 1

根据积分的定义可知原式可以转化为如下定积分
lim(n→∞) (1/n) Σ(i=1→n) (1 + i/n)^(1/3)
= ∫[1→2] x^(1/3) dx
= x^(1 + 1/3)/(1 + 1/3) |[1，2]
= (3/4)x^(4/3) |[1，2]
= (3/4)2^(4/3) - (3/4)(1)
= 3/2^(2/3) - 3/4

Answer 2

lim(n→∞) (1/n) Σ(i=1→n) (1 + i/n)^(1/3)
= ∫[1→2] x^(1/3) dx
= x^(1 + 1/3)/(1 + 1/3) |[1→2]
= (3/4)x^(4/3) |[1→2]
= (3/4)2^(4/3) - (3/4)(1)
= 3/2^(2/3) - 3/4

追问

lim(n→∞) (1/n) Σ(i=1→n) (1 + i/n)^(1/3)
= ∫[1→2] x^(1/3) dx
这一步是怎么得到的，望详解

追答

定积分定义：
∫[a→b] ƒ(x) dx = lim[n→∞] (b - a)/n Σ(k=1→n) ƒ[a + k(b - a)/n]