Calculus II · Unit 4A · lesson

The nth-Term Test for Divergence

Concept

Learning objectives

use the necessary condition $a_n\to0$ to rule out convergence and explain why the converse fails.

The nth-Term Test for Divergence

Explanation

Every convergent series must eventually add almost nothing

If the partial sums $s_N$ converge, then consecutive partial sums approach the same limit. Their difference $s_N-s_{N-1}=a_N$ must therefore approach zero. This gives the quickest possible divergence test: if the terms do not approach zero, the series cannot converge.

The test is one-way. Terms approaching zero are necessary but not sufficient because infinitely many tiny positive contributions may still accumulate without bound. The harmonic series is the canonical warning. Writing " $a_n\to0$ , therefore the series converges" is not a small technical error; it confuses a necessary condition with a sufficient one.

Bridge

A convergent total cannot keep receiving large deposits

If a series converges, its partial sums settle near a finite number. The next term is the change from one partial sum to the next, $a_n=s_n-s_{n-1}$ . Two quantities approaching the same limit must have a difference approaching zero, so convergence forces $a_n\to0$ .

This implication is one-way. Small deposits can still accumulate without bound, as the harmonic series demonstrates. The test is therefore a quick divergence detector, not a convergence test: a nonzero or nonexistent term limit ends the problem, while a zero term limit merely tells us to keep investigating.

**Small terms can still build an unbounded total**
The terms 1/n decrease toward zero while the corresponding harmonic partial sums continue to rise.
Small terms can still build an unbounded total. Compare shrinking harmonic terms with growing harmonic partial sums.

Proof idea

The theorem is a subtraction of two nearby partial sums

If $s_n\to S$ , then also $s_{n-1}\to S$ . Hence

a_n=s_n-s_{n-1}\to S-S=0.

The contrapositive gives the divergence test.

Concept

nth-term test

\lim_{n\to\infty}a_n\ne0

or the limit does not exist, then $\sum a_n$ diverges. If $a_n\to0$ , the test is inconclusive.

Guided walkthrough

Divergence hidden by notation

For

\sum_{n=1}^{\infty}\frac{n}{n+1},

the terms approach $1$ , not zero. Therefore the series diverges immediately. No comparison or ratio test is needed.

Worked example

Use the term test before doing anything elaborate

Consider

\sum_{n=1}^{\infty}\frac{3n^2+1}{2n^2-5}.

The terms satisfy

\lim_{n\to\infty}\frac{3n^2+1}{2n^2-5}=\frac32\ne0.

Therefore the series diverges immediately. No comparison, ratio test, or partial-fraction work is needed.

Common mistake

Terms approaching zero do not prove convergence

The statement $a_n\to0$ is necessary, not sufficient. The harmonic series has terms approaching zero and still diverges.

Interactive checku4a-nth_term_divergence_test-01

Does $\sum_{n=1}^{\infty}\frac{n}{n+1}$ converge or diverge?

Your work stays on this device. No account or AI grader is used.

Show hint

First examine the limit of the term.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Apply the test to $\sum (-1)^n$ .

Exercise

Explain why the test is inconclusive for $\sum1/n$ .

Exercise

Construct a divergent series whose terms approach zero very rapidly for long stretches.

Exercise

Prove the test using $a_N=s_N-s_{N-1}$ .

After the explanation

Use the section idea

Reading lens

Build every infinite sum from finite partial sums, and expose geometric or telescoping structure before taking a limit.

Mental model

A series converges exactly when its sequence of partial sums approaches a finite value.

Decision

Check the term limit first, then look for an exact partial-sum pattern before selecting a comparison test.

Common trap

Concluding convergence from terms approaching zero or canceling inside an unwritten infinite expression.

Check yourself

Can you write the relevant finite partial sum and identify which terms survive?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.