User:Alanonala



\varphi\left(e^{-6\pi}\right) = \frac{\sqrt[3]{3\sqrt{2}+3\sqrt[4]{3}+2\sqrt{3}-\sqrt[4]{27}+\sqrt[4]{1728}-4}\cdot \sqrt[8]{243{\pi}^2}}{6\sqrt[6]{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma(\frac{3}{4})}} $$

\sum_{n=1}^\infty \frac{2}{n(5n-3)} = \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right) $$


 * $$\sum_{n=1}^\infty \frac{2}{n(5n-3)} = $$

Here is a very interesting continued fraction expansion for pi/2 :

\pi/2=1+\cfrac{1}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\cfrac{1}{1/5+\cfrac{1}{1/6+\cfrac{1}{1/7+\cfrac{1}{1/8+\cfrac{1}{1/9+\cfrac{1}{1/10+\cfrac{1}{1/11+\cfrac{1}{1/12+\cfrac{1}{1/13+\cfrac{1}{1/14+\cfrac{1}{1/15+\ddots}}}}}}}}}}}}}}} $$ It appears to converge extremely slowly.