Take the 2-minute tour ×  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.  share|improve this question Every
real number is the sum of countably many rational numbers. – 
Did Mar 4 '12 at 9:47 Properties that are preserved under a
finite number of operations are not necessarily preserved under
an infinite number of them. –  Rahul Mar 4 '12 at 9:57 Real
numbers have the property that the bounded infinite sum of
positive real numbers is a real number because of completeness.
This is not a property of rational numbers, so I would
intuitively expect some bounded infinite sums of positive
rational numbers not to be a rational number. –  Henry Mar 4 '12
at 10:15 because real number are closure of rational numbers. – 
quartz Mar 4 '12 at 12:12 Another example to illustrate Rahul's
point: the product of infinite positive reals can yield zero:
$\prod_{n=1}^\infty 1/n = 0$ or more simply $\prod_{