好きな式集1　(級数積分極限)

$$\sum_{n=0}^{\infty}\dfrac{(3n)!(2n)!}{(n!)^{5}3^{3n}2^{2n}}=\dfrac{9 \Gamma\bigl(\frac{1}{3}\bigl)^{6}}{16\sqrt[3]{4}\,\pi^{4}}$$
$$\sum_{n=0}^{\infty}\dfrac{(_{2n}C_n)^2}{(2n+1)16^n}=\dfrac{4G}{\pi}$$
$$\sum_{n=1}^{\infty}\dfrac{_{2n}C_n}{n4^n}=2\log{2}$$
$$\sum_{n=0}^{\infty}\dfrac{_{2n}C_n}{(n+1)4^n}=2$$
$$\sum_{n=0}^{\infty}\dfrac{_{2n}C_n}{(n+2)4^n}=\dfrac{4}{3}$$
$$\sum_{n=0}^{\infty}\dfrac{_{2n}C_n}{(2n+1)4^n}=\dfrac{\pi}{2}$$
$$\sum_{n=0}^{\infty}\dfrac{_{2n}C_n}{(2n+1)^24^n}=\dfrac{\pi}{2}\log{2}$$
$$\sum_{n=0}^{\infty}\dfrac{_{2n}C_n}{(2n+1)^3 4^n}=\dfrac{\pi^3}{48}+\dfrac{\pi}{4}(\log{2})^2$$
$$\sum_{n=0}^{\infty}\dfrac{1}{(n!)^2}=\dfrac{1}{\pi}\int_{0}^{\pi}e^{2\cos{x}}\,dx$$

$$\int_0^1\sqrt[4]{1-x^4}dx=-\sqrt{\dfrac{\pi}{2}}\dfrac{{\Gamma\bigl(\frac{1}{4}\bigl)}}{{\Gamma\bigl(-\frac{1}{4}\bigl)}}$$
$$\int_{0}^{\infty}\dfrac{x}{\sinh{x}}dx=\dfrac{\pi^2}{4}$$
$$\int_{0}^{\infty}e^{-x^2}\cos{x^2}dx=\dfrac{\sqrt{\pi(\sqrt{2}+1})}{4}$$
$$\int_{0}^{\infty}e^{-x^2}\sin{x^2}dx=\dfrac{\sqrt{\pi(\sqrt{2}-1})}{4}$$
$$\int_{0}^{\frac{\pi}{2}}\dfrac{x^2}{\sin{x}}dx=2\pi G-\dfrac{7\zeta(3)}{2}$$