There are many famous examples demonstrating how adding rational numbers infinite times can give us non rational numbers notably the Basel problem:
$$\sum_{n=0}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$$
but on the other hand, in basic algebra course we are introduced that
$$ (\mathbb{Q}, + ) $$
is an infinite group under addition.
But how do I make sense of these two separate results? On one hand we add a "bunch" of rational numbers to get numbers outside $\mathbb{Q}$, but on the other hand under the group operation all elements in the set should give us elements within that set.
I suspect this has something to do with infinity.
A series of the form $\sum_{n\ge1}n^{-2}$ is a limit of partial sums - but each $\sum_{n=1}^Nn^{-2}$ is a rational number.
Groups, with no additional topological structure, do not need to worry about limits. All you need is that $a+b$ is rational if $a,b$ are rational, and this is true. "Infinite group" means a group with infinitely many elements, it does not mean the group action can be composed infinitely many times.