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)

更新:

咁即係呢個點計?

-3 ∫ x^3/(1+x^3) dx [from 0 to 1]

3 個解答

評分
  • 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:

    圖片參考:http://i212.photobucket.com/albums/cc82/doraemonpa...

    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

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