In the 10 tests of convergence/divergent (that I know), them being,
by defn, integral test, div test, comparison test, limit comparison test, gs test, alternating series test, p-series test, root test, ratio test
the only ones that can tell me what the series converges to is by defn and the gs test.
The other ones can tell me if the series converges or diverges.
But I've always wondered, why do we care whether it converges, if we cannot exactly figure out what it converges to? Why is it helpful to know only the behaviour and not the actual value?