Back to the pi question:
I don't see where the theta comes from. There are no angles in this problem; it's 1-dimensional.
I see the side note that (1-x)/(1+x) is approximately equal to cos theta, and he's calling that a Taylor Series of cos... but...
A) that doesn't look like the Taylor series expansion of cosine.
B) He never defines theta, so it's hard to tell what he's even saying.

A) Check out google images for cosine taylor expansion.
Taylor Series for cos at very small angles is just to set cos(theta) = 1.
At slightly larger angles, we'd say cos(theta) = 1 - (1/2)*(theta)^2
We keep adding higher even powers of theta as we add more terms. We never add a theta to the denominator.
So ... obv. he's a doctorate and I'm not, but I don't see how that's a Taylor Series expansion of cosine.

B)
What the hell does it mean to say f(x) = g(theta) unless you define theta. You don't get to arbitrarily choose any old inputs when you do a series expansion. The whole point is that you're using the same inputs in a function that, while it's less precise, is easier to work with. You gotta at least state how to transform x into theta when you tell me how g(theta) is identical to f(x).