Harmonic Series and p-Series
Use harmonic and p-series benchmarks with the correct exponent and strict convergence threshold.
College Board Topic 10.5 · CED p. 185 (PDF p. 190) · CED effective Fall 2020; project course outline 2026–27.
A p-series is . It converges exactly when . For p>0, the integral test on x⁻ᵖ establishes the dividing line: the improper integral has a finite limit for p>1 and diverges for 0<p≤1. For p≤0, the terms fail to tend to zero, so divergence is even more immediate.
The harmonic series has p=1 and diverges, slowly but without a finite total. Its small terms do not rescue it. Grouping positive terms into blocks of doubling length also reveals at least a fixed positive contribution in each block. The alternating harmonic series behaves differently; its convergence uses the alternating-series test in Topic 10.7.
Multiplying a series by a fixed nonzero constant does not change its convergence. Changing finitely many terms or the starting index also does not change it. But adding a logarithm to the denominator changes the model; a series involving n and ln n is not automatically a p-series. Use a suitable integral or comparison.
Connect to the retained harmonic-series lesson. Treat the familiar p-series as benchmarks for later comparisons, not as a label for every expression containing a power.
Let's work through it
- Classify . Why: Identify p=3/2.
- Since , the series converges. Why: The p-series threshold is strictly greater than 1.
- By contrast, diverges. Why: At p=1, the corresponding logarithmic integral grows without bound.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
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Retrieval 1
Worked answer and reason
It is a p-series with p=1/2≤1.
Retrieval 2
Worked answer and reason
A finite change cannot alter an infinite tail’s divergence.
Catch the tempting mistake
- Every positive p gives convergence.
- The correct threshold is p greater than 1. Exponents greater than zero and at most 1 still diverge.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 10 problem 6 · No calculator · verbal
- Unit 10 problem 9 · No calculator · analytic
- Unit 10 problem 10 · No calculator · analytic
- Unit 10 problem 14 · No calculator · verbal
- Unit 10 problem 16 · No calculator · analytic
For the teacher
- Give students examples with radicals, reciprocal powers and scalar factors, so p is identified from the actual denominator exponent.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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p-series
A p-series has terms 1/nᵖ for positive integers n. It converges exactly when p>1. For example, the p-series with p=2 converges, while p=1 gives the divergent harmonic series. The boundary p=1 belongs to the divergent cases.
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Integral test
The integral test compares a positive-term series with an improper integral of a matching function. The function must be continuous, positive, and decreasing for all sufficiently large inputs, with f(n) equal to the series terms. For example, applying it to 1/n² uses the convergent integral of 1/x² over [1,∞). The test determines convergence together, but the integral’s value is generally not the series sum.
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Harmonic series
The harmonic series is 1+1/2+1/3+1/4+⋯. It diverges even though its individual terms approach 0. For example, grouping terms from 1/2 through 1/3+1/4 and then 1/5+⋯+1/8 shows repeatedly added blocks of at least 1/2. It is a standard comparison that separates a necessary term condition from actual convergence.
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Convergence of a series
A series converges when the sequence of its partial sums approaches a finite limit. It diverges when there is no such finite limit. For example, 1/2+1/4+1/8+⋯ converges to 1, while the harmonic series diverges. Convergence of the individual terms to 0 is necessary but does not by itself establish convergence of their sum.
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Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
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Index of a sequence or sum
An index identifies the position of a term in a sequence or finite or infinite sum. The starting index determines which term is included first. For example, a sum indexed by n=0 starts with the term obtained by substituting 0 into its term formula. Reindexing is valid only when the formula and bounds change together so that the same terms are retained.