Comparison Tests for Convergence
Choose a positive comparison with the useful inequality direction or a finite positive ratio limit.
College Board Topic 10.6 · CED p. 186 (PDF p. 191) · CED effective Fall 2020; project course outline 2026–27.
Direct comparison works with nonnegative terms. If on a tail and converges, then converges too. If and diverges, then diverges. Being smaller than a divergent series, or larger than a convergent one, does not settle the question.
Limit comparison can avoid awkward inequalities. For positive terms aₙ and bₙ, if with , the two series share convergence behavior. State the benchmark and the limit. Limits of 0 or infinity require a one-sided argument and do not satisfy this two-way version of the theorem.
Use dominant powers to choose a benchmark, then justify it. For , choose bₙ=1/n. The ratio tends to 3, which is finite and positive, so the series diverges with the harmonic series. The phrase looks like 1/n is a good idea starter, but the limit turns it into a proof.
Read the retained comparison lesson. A useful classroom habit is to write the required inequality direction before attempting the algebra.
Let's work through it
- For , use . Why: Adding 3 enlarges the denominator of a positive reciprocal.
- converges. Why: It is a p-series with p=2>1.
- The given series converges by direct comparison. Why: It is no larger than a convergent positive benchmark.
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
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Worked answer and reason
An upper bound with a divergent sum is inconclusive.
Retrieval 2
Worked answer and reason
A finite positive ratio means the terms have comparable scale.
Catch the tempting mistake
- Any inequality with a known series settles convergence.
- The inequality must face the useful direction: bounded above by a convergent sum or below by a divergent sum.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 10 problem 11 · No calculator · analytic
- Unit 10 problem 12 · No calculator · analytic
For the teacher
- Sort proposed comparisons into valid and inconclusive piles before calculating any new limits.
Glossary and deeper reading · 4 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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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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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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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.