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(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.

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    $\begingroup$ By the same logic (just replacing "rational" by "finite", which makes no difference to your argument): (1) the sum of two finite numbers is a finite number. Therefore (2) the sum of an infinite number of finite numbers should be a finite number. $\endgroup$ – TonyK Nov 13 '12 at 20:37
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    $\begingroup$ Let $x$ be a real number, for simplicity between $0$ and $1$. Let $x$ have decimal expansion $0.a_1a_2a_3\dots$. Then $x=\frac{a_1}{10}+\frac{a_2}{10^2}+\frac{a_3}{10^3}+\cdots$. So $x$ is the sum of a series with rational terms. $\endgroup$ – André Nicolas Nov 13 '12 at 20:43
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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 )

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