(1) The sum of two rational numbers is a rational number.
(2) The series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$ is irrational.
The equation (2) is repeating (1) infinitely many times. So, why (2) is not rational? I get that it is the infinity messing things up, but cannot figure out why.