Concept preview

What you will be able to do

1. 01State the nth-term test for divergence: if $\lim_{n\to\infty} a_n \ne 0$ (or the limit does not exist), then the series $\sum a_n$ diverges.
2. 02Explain the one-directional logic of the test — why $\lim_{n\to\infty} a_n = 0$ is necessary for convergence but is NOT sufficient, using the harmonic series as the decisive counterexample.
3. 03Use the test as a fast first screen on a series, correctly reporting "diverges" or "inconclusive" and never mistakenly concluding convergence from shrinking terms.

A series $\sum_{n=1}^{\infty} a_n$ converges only if its sequence of partial sums $S_N = a_1 + a_2 + \cdots + a_N$ approaches a finite limit as $N\to\infty$. Now think about what each new term $a_N$ contributes: $a_N = S_N - S_{N-1}$. If the partial sums are homing in on a single number $S$, then both $S_N$ and $S_{N-1}$ approach that same $S$, and their difference $a_N$ must approach $S - S = 0$. In other words, convergence FORCES the terms to die out. Turning that statement around gives the test.