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10.3 The nth Term Test for Divergence

Syllabus
2020
Topic
10.3
Level
—

If the Terms Do Not Approach Zero, the Series Diverges

If ∑an\sum a_n converges, adding the next term must eventually make almost no change to its partial sum. Because an=Sn−Sn−1a_n=S_n-S_{n-1}, convergence of the partial sums forces an→0a_n\to0. Therefore any failure of the terms to approach zero proves divergence.

\lim_{n\to\infty}a_n\ne0\text{ or does not exist}\Rightarrow\sum_{n=1}^{\infty}a_n\text{ diverges};\qquad \lim_{n\to\infty}a_n=0\Rightarrow\text{inconclusive}

Example 1: for ∑n=1∞n/(n+1)\sum_{n=1}^{\infty}n/(n+1),
lim⁡n→∞nn+1=1≠0.\lim_{n\to\infty}\frac{n}{n+1}=1\ne0.
The terms keep contributing about 11, so the nth term test proves that the series diverges. No other convergence test is needed.

Example 2: for ∑n=1∞1/n\sum_{n=1}^{\infty}1/n, the term limit is 00. The nth term test is inconclusive: it does not say the series converges. A different test is required to determine its behavior.

The test is designed to prove divergence, not convergence. The statement “a convergent series has an→0a_n\to0” is true, but its converse is false: terms approaching zero is necessary, not sufficient, for the accumulated sum to approach a finite value.