What exactly prevents me (or, if I'm just paranoid, doesn't) from making operations such as inverses on summations?
For instance, can I say that
$$({\sum_{n=1}^{\infty}n})^{-1}=(-\frac{1}{12})^{-1}=\sum_{n=1}^{\infty}\frac{1}{n}=-12?$$
I can already tell this may be wrong, but either way, I would like to know exactly why and where I may look up rules and axioms of summation in case my mind ever comes across something like this again. I thank you in advance.
Well, for one, exponentiation (of which inversion is a type) doesn't distribute over addition, so this won't work even for finite sums; e.g., $(2+2)^{-1} \neq 2^{-1}+2^{-1}$. You want to make sure something works for finite sums before you try it on infinite sums.