The series 1+2+3+4+.... diverges, right? In fact, it "obviously" diverges, right? Read to the "contents" box of the wikipedia article. If your mind isn't blown, check out this video. By the way, the details of this logic is something that would be revealed in either a complex analysis course (typically a 300-level course for math majors), or an analytic number theory course (which, sadly yet honestly, you probably wouldn't see before grad school).
One of the classic examples from the sequences and series portion of calc 2 is the harmonic series. I mentioned that it is also well known that the sum of (1/p) for p prime also diverges. Here is a link to a 2015 paper (while very technical, and while perhaps a challenging read, this is in a well-known, predominantly expository math journal) on "how" that latter series diverges.
Terry Tao just announced a proof of the Erdos discrepancy problem.