sequences and series - why does math "break" when it comes to certain topics?

For example,

$1 + 2 + 3 + ... = -1/12$

or $2 + 4 + 8 + ... = -1$

What caused it to break? Are there some properties of real numbers that do this? Sorry if it's too vague because I was looking at group theory and there were things that didn't hold.

Another thing is why do some problems become ill-posed? For example, Wilkinson Polynomial, why do roots break by having too many multiplication of binomials?