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Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?
For an education assignment, I would like a selection of various results, identities, or fomulae in which $\pi = 3.14159...$ appears. In particular, results which are not immediately obvious as being totally geometric, as the definition of $\pi$ is, for example. So far I have recalled Euler's $$\sum_{n=1}^{\infty} n^{-2} = \frac{\pi^2}{6},$$ and identities involving the Gamma function, Stirling's formula and its use in probability. I wish to create an excercise whereby students may evaluate the pros and cons of universally replacing $\pi $ with $\frac{\tau }{2}$ throughout mathematical literature and textbooks, but also learn mathematics. Question 26918 is similar.
