ウォリス積分を含む極限
ウォリス積分を含む極限値
(1)
\[ \lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\sqrt{\frac{\pi}{2}} \](2)
\[ \lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\cos^{n}\theta d\theta=\sqrt{\frac{\pi}{2}} \](1)
\begin{align*} \lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta & =\frac{1}{2}\lim_{n\rightarrow\infty}\sqrt{n}B\left(\frac{n+1}{2},\frac{1}{2}\right)\\ & =\frac{1}{2}\lim_{n\rightarrow\infty}\sqrt{n}\frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)}\\ & =\frac{\sqrt{\pi}}{2}\lim_{n\rightarrow\infty}\sqrt{\sqrt{n}\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)}\sqrt{n+1}\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+3}{2}\right)}}\\ & =\frac{\sqrt{\pi}}{2}\lim_{n\rightarrow\infty}\sqrt{\sqrt{n}\sqrt{n+1}\frac{2}{n+1}}\\ & =\sqrt{\frac{\pi}{2}}\lim_{n\rightarrow\infty}\sqrt{\frac{1}{\sqrt{1+\frac{1}{n}}}}\\ & =\sqrt{\frac{\pi}{2}} \end{align*}(2)
\begin{align*} \lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\cos^{n}\theta d\theta & =\lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}\left(\frac{\pi}{2}-\theta\right)d\theta\\ & =\lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}tdt\qquad,\qquad t=-\theta+\frac{\pi}{2}\\ & =\sqrt{\frac{\pi}{2}} \end{align*}
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数列の極限での大小関係
\[
a_{n}<b_{n}\Rightarrow a\leq b
\]
合成関数の導関数・偏導関数
\[
\frac{df}{dt}=\sum_{k=1}^{n}\frac{\partial f}{\partial x_{k}}\frac{dx_{k}}{dt}
\]
偏微分の順序交換(シュワルツの定理)
\[
\frac{\partial^{2}f\left(x,y\right)}{\partial x\partial y}=\frac{\partial^{2}f\left(x,y\right)}{\partial y\partial x}
\]
C1級・全微分可能・偏微分可能・連続の関係
\[
C^{1}\text{級}\Rightarrow\text{全微分可能}\Rightarrow\text{偏微分可能}
\]