jlggljct 發問於 科學及數學數學 · 1 十年前

# definite integration

lim (n->infinity) {(n^3+1)(n^3+2^3)(n^3+3^3)...(n^3+n^3) / n^(3n)}^(1/n)

-3 ∫ x^3/(1+x^3) dx [from 0 to 1]

### 3 個解答

• 1 十年前
最愛解答

不是嗎，myisland8132，計算∫ ln(1+x^3) dx也要淪落到用數值積分？

http://integrals.wolfram.com/index.jsp?expr=ln%281...

2010-12-18 02:40:00 補充：

參考資料：

my maths knowledge

• 1 十年前

∫ ln(1+x^3) dx [from 0 to 1] = 0.200094 not -2.8

• 1 十年前

lim (n->infinity) {(n^3+1)(n^3+2^3)(n^3+3^3)...(n^3+n^3) / n^(3n)}^(1/n)

=lim (n->infinity) {(1+(1/n)^3)(1+(2/n)^3)(1+(3/n)^3)...(1+(n/n)^3)}^(1/n)

Now let y=lim (n->infinity) {(1+(1/n)^3)(1+(2/n)^3)(1+(3/n)^3)...(1+(n/n)^3)}^(1/n), then lny

=lim (n->infinity) (1/n) {ln(1+(1/n)^3)+ln(1+(2/n)^3)+ln(1+(3/n)^3)...ln(1+(n/n)^3)}

= ∫ ln(1+x^3) dx [from 0 to 1]

=xln(1+x^3)|[0,1]- ∫ 3x^3/(1+x^3) dx [from 0 to 1]

=ln2-3 ∫ x^3/(1+x^3) dx [from 0 to 1]

There is no closed form of the last integral. By approximation, its value is around 1.16435. So, lny=-2.8 and the value of the original expression is e^(-2.8) = 0.06082