I didn’t create this blog until after Pi day, so unfortunately I’m a bit late. I celebrate Pi day by deriving the same series expression for pi every year. I’m not sure who to attribute this particular method to, but I definitely learned in from some old calculus book once upon a time. It’s as simple as calculating two different ways and equating the results. First up, we can make the substitution and simplify yielding . On the other hand, if then converges (uniformly on all intervals ) to , and so may proceed with
. Each , and so putting it all together we’re left with
And there it is!