From Emanuel Kowalski, quoting what is apparently folklore: "The only 'truly' divergent series is the harmonic series."
The idea is that one can assign a value somehow to basically any other divergent series; see the link for a more thorough explanation.
(But don't tell the calculus students! They already can't remember the harmonic series is divergent.)
And I usually think of the harmonic series as being "equal" to log n, although of course log n isn't a number. So I amend the folklore, somewhat facetiously: "there are no divergent series."