(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.
The problem is that "addition" and "summing" are two different things.
Summing, as in (2) requires both the act of addition and the act of taking limits.
And we know that taking limits of rational numbers does not preserve rationality.
(there is a sequence converging to $\sqrt{2}$ in $\mathbb{Q}$ for example )