who gets what pi?

The amount of pie Fred will have is $1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} - \ldots$

Consider the derivative of $\tan^{\text{-}1}x$ and its power series expansion:

$\dfrac{d}{dx} \tan^{\text{-}1} x = \dfrac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + x^8 - \ldots \qquad \text{for -}1 < x < 1$

Power series can be integrated term-by-term within their radius of convergence, which gives us

$\tan^{\text{-}1} x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \dfrac{x^9}{9} - \ldots$

Since the terms in the above series are strictly decreasing and converging to zero for - $1 \le x \le 1$ (note the inclusion of the boundary values here) the Alternating Series Test tells us that the series will converge within that range. Finally, letting $x=1$ , we get

$1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} - \ldots = \tan^{\text{-}1} 1 = \boxed{\dfrac{\pi}{4}}$

This problem can be expressed as the solution to $\sum_{n=1}^{\infty} \frac{1}{4n-3} - \frac{1}{4n-1}$ or equivalently, $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n-1}$ This series is known as the Leibniz formula, which has been shown to equal $\frac{\pi}{4}$ . A variety of proofs exist for this formula: here is an elementary proof, and here is Leibniz's original geometric proof.