Can we always establish whether an infinite series converges or diverges? I'm currently working with infinite series for my calculus class, and I'm wondering whether we always (in theory) can establish whether a series is divergent or convergent? Of course, it might be computationally hard, but is there a class of series where we simply lack the tools to determine whether the series converges or diverges?

Every series is convergent or divergent.
However there is no general method to find if any series is indeed convergent or divergent.
There are two reasons.
Some series (most of them) are not expressible in any shortened form to even define them except stating each term, meaning the sum of such series is not known except stating each partial sum, but then you can say nothing about its convergence or divergence.
Extracting the information about convergence means we should be able to extract at least that much about any series. Unfortunately there are series (most of them actually) that you cannot extract any of the global properties. None. Only a couple of first terms and consequently partial sums. Nothing else. Nada, zero, rien, nichts, τίποτα.
For the series whose partial sums are compressible, there is no universal algorithm, so they are recursively undecidable.
The deep reason for this is that divergence and convergence are easily made equivalent to an algorithm halts or does not halt. You can effectively turn any algorithm into a series whose convergence/divergence you want to analyze.
We cannot know which series we can solve by which method. But each particular series that is compressible might (should) have a particular method that would decide if it is convergent or divergent. So each compressible series is likely a world on its own.
So it is triple naked :(
