Infinite Union/Intersection vs Infinite summation I have a question for you. 

It's well known that I can't sum an infinite number of "number". Indeed, the series
$$
\sum_{k=1}^\infty a_k
$$
is defined as a Limit. 

However, I can make the infinite union/intersection of sets. Someone can explain me why ?
Take this as the definition of infinite unions and intersections:

$x \in A_1 \cup A_2 \cup A_3 \dots $ if and only if $x \in A_i$ for at least one $i \in \mathbb N$.

$x \in A_1 \cap A_2 \cap A_3 \dots $ if and only if $x \in A_i$ for all $i \in \mathbb N$.

This makes perfect sense as it is. There is no need for a limiting procedure.

(In fact, there is even no need to assume that there are countably many $A_i$'s...)

The reason that sum of a series is different is that we define it inductively. Say we're trying to work out
$$ a_1 + a_2 + a_3 + a_4\dots $$
Well, let $S_i$ be the sum of the first $i$ terms. Then you define
$$ S_1 = a_1, \ \ \ \ S_2 = S_1+ a_2, \ \ \ \ S_3 = S_2 + a_3, \ \ \ \ S_4 = S_3 + a_4, \dots $$

But the problem with this is that, no matter how many steps you do, you will never reach $S_\infty$. This is why summing infinite series requires a special procedure.
The countable intersection and union are also limits. $$\bigcup_{n \in \mathbb{N}} A_n = \lim_{n \to \infty} \bigcup_{i=1}^n A_n$$

Set operations can be defined without reference to the size of the domain of "summation": $$\bigcup A := \{ x: x \in X \in A \}$$
$$\bigcap A := \{x : (\forall X \in A)(x \in x) \}$$
(no countability needed).

Note that infinite sums are only really defined when the domain of summation is countable (see this question). So infinite sums are really very restricted objects compared to sets.