Calculus II · Unit 4A · lesson
The nth-Term Test for Divergence
Use the nth-term test correctly as a one-way divergence test and understand why terms approaching zero do not guarantee convergence.
Section overview
Infinite series and foundational examplesWhat this section is building
Use the nth-term test correctly as a one-way divergence test and understand why terms approaching zero do not guarantee convergence.
A series converges exactly when its sequence of partial sums approaches a finite value.
Check the term limit first, then look for an exact partial-sum pattern before selecting a comparison test.
Concluding convergence from terms approaching zero or canceling inside an unwritten infinite expression.
Learning objectives
use the necessary condition to rule out convergence and explain why the converse fails.
The nth-Term Test for Divergence
Every convergent series must eventually add almost nothing
If the partial sums converge, then consecutive partial sums approach the same limit. Their difference 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 ", therefore the series converges" is not a small technical error; it confuses a necessary condition with a sufficient one.
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, . Two quantities approaching the same limit must have a difference approaching zero, so convergence forces .
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.
Read this graph as text
Small terms can still build an unbounded total. The terms 1/n decrease toward zero while the corresponding harmonic partial sums continue to rise. Compare shrinking harmonic terms with growing harmonic partial sums. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in small terms can still build an unbounded total; color is never the only cue.
Why it matters: Compare shrinking harmonic terms with growing harmonic partial sums.
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.
The theorem is a subtraction of two nearby partial sums
If , then also . Hence
The contrapositive gives the divergence test.
nth-term test
If
or the limit does not exist, then diverges. If , the test is inconclusive.
Divergence hidden by notation
For
the terms approach , not zero. Therefore the series diverges immediately. No comparison or ratio test is needed.
Use the term test before doing anything elaborate
Consider
The terms satisfy
Therefore the series diverges immediately. No comparison, ratio test, or partial-fraction work is needed.
Terms approaching zero do not prove convergence
The statement is necessary, not sufficient. The harmonic series has terms approaching zero and still diverges.
u4a-nth_term_divergence_test-01Does converge or diverge?
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First examine the limit of the term.
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Apply the test to .
Explain why the test is inconclusive for .
Construct a divergent series whose terms approach zero very rapidly for long stretches.
Prove the test using .
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