From one of my past Professors, Dr. Paul Loya's, book:
\(\displaystyle \frac{2}{\pi} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}} \dots\)
\(\displaystyle \frac{\pi}{4} = \frac{1}{1} - \frac{1}{3} +\frac{1}{5} -\frac{1}{7} + \dots\)
\(\displaystyle \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} +\frac{1}{3^2} +\frac{1}{4^2} + \dots\)
\(\displaystyle \frac{\pi^4}{90} = \frac{1}{1^4} + \frac{1}{2^4} +\frac{1}{3^4} +\frac{1}{4^4} + \dots\)
\(\displaystyle \frac{\pi}{2} = \frac{1}{1} \cdot\frac{2}{1} \cdot\frac{2}{3} \cdot\frac{4}{3} \cdot\frac{4}{5} \cdot\frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \dots\)
\(\displaystyle \Phi = 1+\frac{1}{1+\frac{1}{1+ \dots}}\)
\(\displaystyle \Phi = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\)
\(\displaystyle e = 2 + \frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\dots}}}}\)
There are a few more and references to the finders of these as well if you're interested.
\(\displaystyle \frac{2}{\pi} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}} \dots\)
\(\displaystyle \frac{\pi}{4} = \frac{1}{1} - \frac{1}{3} +\frac{1}{5} -\frac{1}{7} + \dots\)
\(\displaystyle \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} +\frac{1}{3^2} +\frac{1}{4^2} + \dots\)
\(\displaystyle \frac{\pi^4}{90} = \frac{1}{1^4} + \frac{1}{2^4} +\frac{1}{3^4} +\frac{1}{4^4} + \dots\)
\(\displaystyle \frac{\pi}{2} = \frac{1}{1} \cdot\frac{2}{1} \cdot\frac{2}{3} \cdot\frac{4}{3} \cdot\frac{4}{5} \cdot\frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \dots\)
\(\displaystyle \Phi = 1+\frac{1}{1+\frac{1}{1+ \dots}}\)
\(\displaystyle \Phi = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\)
\(\displaystyle e = 2 + \frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\dots}}}}\)
There are a few more and references to the finders of these as well if you're interested.
