カテゴリー: 解析学

\[ F\left(x_{1},\cdots,x_{n},\lambda_{1,}\cdots,\lambda_{m}\right)=f\left(x_{1},\cdots,x_{n}\right)-\sum_{k=1}^{m}\lambda_{k}g_{k}\left(x_{1},\cdots,x_{n}\right) \]

\[ \lim_{n\rightarrow\infty}\frac{x^{n}}{n!}=0 \]

\[ \sum_{k=0}^{\infty}C^{-1}\left(2k,k\right)=\frac{4}{3}+\frac{2\sqrt{3}\pi}{27} \]

\[ \lim_{n\rightarrow\infty}\sqrt{n}4^{n}B(n,n)=2\sqrt{\pi} \]

\[ \lim_{n\rightarrow\infty}\sqrt{n}\frac{\Gamma\left(n\right)}{\Gamma\left(n+\frac{1}{2}\right)}=1 \]

\[ \lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\sqrt{\frac{\pi}{2}} \]

\[ \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta d\theta=\frac{C(2m,m)}{4^{m}}\frac{\pi}{2} \]

\[ \int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\int_{0}^{\frac{\pi}{2}}\cos^{n}\theta d\theta \]

\[ \int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta \]

\[ \prod_{k=1}^{\infty}\left(\frac{(2k)^{2}}{(2k-1)(2k+1)}\right)=\frac{\pi}{2} \]