n-th term test
The n-th term test is the first thing to check, and its only honest job is to spot losers: if the individual terms of a series do not shrink to zero, the series cannot possibly converge. It is a quick disqualifier, not a passport to convergence.
Precisely: if sum a_n converges, then a_n -> 0. Taking the contrapositive gives the usable form — if a_n does not tend to 0 (or tends to a nonzero limit, or has no limit), then sum a_n diverges. The proof is one line: a_n = s_n - s_{n-1}, and if s_n -> S then a_n -> S - S = 0.
The crucial caveat, repeated to every student: the converse is false. The condition a_n -> 0 is necessary but not sufficient for convergence. The harmonic series sum 1/n has terms tending to 0 yet diverges. So when the test shows a_n -> 0, it tells you nothing — the test is inconclusive, and you must reach for a stronger tool such as the comparison, ratio, root, or integral test.
For sum n/(n+1), the terms n/(n+1) -> 1, not 0, so the series diverges. But for sum 1/n the terms -> 0 and the test is silent, even though that series diverges too.
The test detects divergence but never certifies convergence.