TITLE: How can adding an infinite number of rationals yield an irrational number?
QUESTION [94 upvotes]: For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.

REPLY [4 votes]: For example look at 
$$e = \sum_{k=0}^{\infty}{\frac{1}{k!}} $$