Why a Nonzero Term Limit Proves a Series Diverges

Why Inspect the Terms of a Series?

A series is defined through the behavior of its partial sums, not by adding infinitely many terms all at once. Yet the individual terms can provide a quick way to rule out convergence. If the terms fail to approach zero, the partial sums cannot settle toward a finite value.

This is a one-way test. A nonzero term limit, or a term sequence with no limit, proves divergence. A zero term limit does not prove convergence; it only leaves that possibility open.

Series and Partial Sums

For a sequence of real numbers \(a_1,a_2,a_3,\dots\), the infinite series \(\sum_{n=1}^{\infty}a_n\) is defined by the sequence of partial sums

The series converges if the sequence of partial sums \(S_N\) approaches a finite real number as \(N\to\infty\). If the partial sums do not approach a finite limit, the series diverges. The terms \(a_n\) and the partial sums \(S_N\) are related, but they are not the same sequence.

That distinction is central to the term test. The test does not calculate the sum. It asks whether the terms have a behavior that makes convergence impossible.

The Necessary Condition for Convergence

Each term after the first is the difference between two consecutive partial sums. In particular, for \(n\ge 2\),

If the series converges, then both sequences \(S_n\) and \(S_{n-1}\) approach the same limit. Their difference must therefore approach zero. The following derivation makes that implication explicit.

This theorem is a necessary-condition test: every convergent series must pass the test \(\lim a_n=0\). Its contrapositive gives a useful way to prove divergence. If the terms have a nonzero limit, have no limit, or otherwise fail to approach zero, the series cannot converge.

Applying the nth-Term Divergence Test

The test is also called the nth-term test for divergence. It applies to series with positive, negative, or alternating terms; no assumption that the terms are positive is needed. First find the limit of \(a_n\), the general term. Then compare the result with the necessary condition.

Why a Zero Term Limit Is Not Enough

The theorem gives an implication in one direction: if a series converges, then its terms approach zero. It does not say that terms approaching zero force the series to converge. In logic, reversing an implication without proof is invalid. To see the issue concretely, consider the harmonic series.

A Common Mistake and Its Correction

A common incorrect argument is: “For \(\sum_{n=1}^{\infty}\frac{1}{n}\), the terms approach zero, so the series converges.” The calculation \(\lim_{n\to\infty}\frac1n=0\) is correct, but the conclusion does not follow. It treats a necessary condition as though it were sufficient.

The correction is to state exactly what the term limit tells you: a nonzero limit or failure to approach zero proves divergence; a zero limit gives no conclusion by itself. For the harmonic series, the block estimate above supplies the additional argument needed to prove divergence.

Using the Test Effectively

The nth-term divergence test is often most valuable before using a more involved method. It can quickly eliminate a series from consideration when its terms do not approach zero. But if the term limit is zero, stop short of claiming convergence: another argument is needed to decide the series.

• If \(\lim a_n=L\ne 0\), conclude that \(\sum a_n\) diverges.
• If the terms have no limit and do not approach zero, conclude that the series diverges.
• If \(\lim a_n=0\), record that the test is inconclusive; the series may converge or diverge.

The reason for this asymmetry is built into the definition of a series: convergence concerns the partial sums. When those sums converge, consecutive sums become close, forcing their difference \(a_n\) to approach zero. But differences approaching zero alone do not guarantee that the sums settle to a finite value.