Some Beautiful Formulas:

[daon]

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.

 

[galactus]

Daon, have you tried deriving them?.

 

[Deleted member 4993]

I think the second one can be derived from Taylor series expansion of tan[sup:29ehdesw]-1[/sup:29ehdesw](1)

 

[daon]

Daon, have you tried deriving them?.

No, I haven't. I'm sure I was required to prove at least some of these though... either by set supremum or induction.