数列 \(\{ a _ n \} \ ( a _ n \gt 0 )\) を次の規則によって定める： \[ a _ 1 = 1 \ : \ \displaystyle\int _ {a _ n}^{a _ {n+1}} \dfrac{dx}{\sqrt[3]{x}} = 1 \quad ( n = 1, 2, 3, \cdots ) \] 曲線 \(y= \dfrac{1}{\sqrt[3]{x}}\) と, \(x\) 軸および \(2\) 直線 \(x = a _ n\) , \(x = a _ {n+1}\) で囲まれた図形を \(x\) 軸の周りに \(1\) 回転させた回転体の体積を \(V _ x\) とする. このとき, \(\displaystyle\lim _ {n \rightarrow \infty} \sqrt{n} V _ x\) を求めよ.

【 解 答 】

\[\begin{align} \displaystyle\int _ {a _ n}^{a _ {n+1}} \dfrac{dx}{\sqrt[3]{x}} & = \left[ \dfrac{3}{2} x^{\frac{2}{3}} \right] _ {a _ n}^{a _ {n+1}} \\ & = \dfrac{3}{2} \left( {a _ {n+1}}^{\frac{2}{3}} -{a _ n}^{\frac{2}{3}} \right) \ . \end{align}\] なので, 条件より \[\begin{align} \dfrac{3}{2} \left( {a _ {n+1}}^{\frac{2}{3}} -{a _ n}^{\frac{2}{3}} \right) & = 1 \\ \text{∴} \quad {a _ {n+1}}^{\frac{2}{3}} & = {a _ n}^{\frac{2}{3}} +\dfrac{2}{3} \ . \end{align}\] したがって, 数列 \(\left\{ {a _ n}^{\frac{2}{3}} \right\}\) は初項 \({a _ 1}^{\frac{2}{3}} = 1\) , 公差 \(\dfrac{2}{3}\) の等差数列なので \[ {a _ n}^{\frac{2}{3}} = 1 +\dfrac{2}{3} (n-1) = \dfrac{2n-1}{3} \quad ... [1] \ . \] また \[\begin{align} V _ x & = \pi \displaystyle\int _ {a _ n}^{a _ {n+1}} \left( \dfrac{1}{\sqrt[3]{x}} \right)^2 \, dx \\ & = \pi \left[ 3 x^{\frac{1}{3}} \right] _ {a _ n}^{a _ {n+1}} \\ & = 3 \pi \left( {a _ {n+1}}^{\frac{1}{3}} -{a _ n}^{\frac{1}{3}} \right) \ . \end{align}\] これと [1] を用いれば \[\begin{align} \sqrt{n} V _ x & = 3 \pi \sqrt{n} \left( \sqrt{\dfrac{2n+1}{3}} -\sqrt{\dfrac{2n-1}{3}} \right) \\ & = \sqrt{3n} \pi \cdot \dfrac{2}{\sqrt{2n+1} +\sqrt{2n-1}} \\ & = \dfrac{2 \sqrt{3} \pi}{\sqrt{2 +\frac{1}{n}} +\sqrt{2 -\frac{1}{n}}} \\ & \rightarrow \dfrac{2 \sqrt{3} \pi}{2 \sqrt{2}} \quad ( \ n \rightarrow \infty \text{のとき} ) \\ & = \dfrac{\sqrt{6}}{2} \pi \ . \end{align}\] よって \[ \displaystyle\lim _ {n \rightarrow \infty} \sqrt{n} V _ x = \underline{\dfrac{\sqrt{6}}{2} \pi} \ . \]