Topic 10.3 The nth Term Test for Divergence: use the limit of the terms to conclude divergence when the terms do not approach zero (BC).

What this topic is asking

The College Board (Topic 10.3, BC only) gives the $n$-th term test for divergence, the quickest divergence check and the one to try first on any series. It uses a simple necessary condition: if a series converges, its terms must shrink to zero, so terms that do not shrink to zero force divergence.

The test and its one-way logic

Why it can only prove divergence

The test rests on a theorem: if $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 0$. Logically, the test is the contrapositive: if the terms do not go to zero, the series cannot converge, so it diverges. But the original theorem is a one-way implication; its converse ("terms go to zero implies convergence") is false. The decisive counterexample is the harmonic series $\sum\frac{1}{n}$, whose terms $\frac{1}{n}\to 0$ yet whose partial sums grow without bound. This is why $\lim a_n = 0$ leaves you no conclusion: the condition is necessary for convergence but not sufficient. Explaining this distinction is a recurring free-response request, as in the worked exam question above.

A worked divergence conclusion

When the limit does not exist

The test also fires when the terms have no limit at all, not just a nonzero one. For $\sum (-1)^n$, the terms alternate $-1, 1, -1, 1, \ldots$ and never settle, so $\lim a_n$ does not exist; the series diverges. Likewise $\sum\cos n$ has terms that oscillate without approaching anything, giving divergence. The unifying statement is that convergence requires $\lim a_n = 0$; any failure of that, whether a nonzero limit or no limit, means divergence. So when you compute the limit of the terms, watch for oscillation as well as nonzero values, and treat both as immediate divergence.

Where it fits in the testing strategy

The $n$-th term test is the first move in any convergence problem because it is cheap and decisive when it applies. If the terms obviously do not tend to zero (a ratio of equal-degree polynomials, a constant, an oscillation), you are done: divergence. If the terms do tend to zero, you have learned only that you need a real test, the integral test, comparison, ratio, alternating-series, or $p$-series, depending on the form. Building the habit of checking $\lim a_n$ first saves time and catches divergent series that a more elaborate test would also catch but more slowly. Just never write "the terms go to zero, so the series converges," which inverts the test illegitimately.