Infinite Union/Intersection vs Infinite summation

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.

First of all, I am not sure why you can't sum an infinite number of terms (other than summing them one by one of course). If $a_i = 0$ for all $i$, then the sum is $0$ ... no limit needed.

But yes, typically we have to use a limit to evaluate the value of the sum.

The difference with sets is that we are not evaluating anything ... we are simply defining what the elements of the union or intersection are.