The nth Term Test for Divergence

A sum of infinitely many numbers can only settle to a finite value if the pieces you're adding eventually become negligible. If a_n does not approach 0, then every term you add keeps contributing a real amount, so the running total either grows without bound or oscillates forever. That's the whole idea behind the nth Term Test for Divergence: compute lim(n→infinity) a_n. If that limit is nonzero, or doesn't exist, the series sum(a_n) diverges. No further work needed.

Here's the catch: this test only ever proves divergence. If lim a_n = 0, you've learned nothing about convergence yet, because terms shrinking to zero is necessary but not sufficient. The harmonic series sum(1/n) is the classic warning: 1/n → 0, yet the partial sums grow without bound and the series diverges anyway. So a zero limit just means 'try another test' — ratio, integral, comparison, whatever fits the series' shape.

This test is usually your first move on any series, because it's fast: find the limit, and if it's not 0, you're done. It sits right before the more delicate tools later in the unit that handle the zero-limit case, like the integral test, comparison tests, and the ratio test, which actually measure how fast the terms shrink rather than just whether they shrink at all.