# Irrationality from infinite rational series

(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.

• 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. – TonyK Nov 13 '12 at 20:37
• 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. – André Nicolas Nov 13 '12 at 20:43

(there is a sequence converging to $\sqrt{2}$ in $\mathbb{Q}$ for example )