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Series Convergence Tests

A free GRE Mathematics note on the convergence toolkit: geometric and p-series benchmarks, comparison, ratio, and root tests, the alternating series test, and the difference between absolute and conditional convergence.

Concise answer

Series classification is benchmark-driven: geometric series and p-series are the two families whose behavior you know outright, comparison tests transfer that knowledge to positive-term lookalikes, ratio and root tests handle factorials and exponentials, and the alternating series test — together with the absolute-versus-conditional distinction — covers sign-alternating series that the absolute-value benchmarks reject.

Definitions

Geometric series
A series with constant ratio between consecutive terms; it converges to $\dfrac{a}{1 - r}$ exactly when the ratio satisfies $|r| < 1$.
p-series
The series $\sum_{n=1}^{\infty} \dfrac{1}{n^p}$; it converges when $p > 1$ and diverges when $p \le 1$.
Absolute convergence
A series $\sum a_n$ converges absolutely when $\sum |a_n|$ converges; absolute convergence implies convergence.
Conditional convergence
A series that converges while the series of its absolute values diverges — convergence that depends on cancellation between positive and negative terms.
Alternating series test
If the terms of an alternating series decrease in absolute value with limit zero, the series converges.

Intuition

Every convergence test is a comparison in disguise: ratio and root tests compare a series against geometric decay, and the p-series boundary at p=1p = 1 marks how fast terms must shrink when there is no cancellation to help. Terms going to zero is necessary but nowhere near sufficient — the harmonic series is the standing counterexample.

Alternation buys extra convergence because consecutive partial sums bracket the limit: each new term overshoots and the next corrects. That is why ∑(−1)n+1/n\sum (-1)^{n+1}/n converges while its absolute-value series diverges — the convergence lives entirely in the cancellation.

Concept walkthrough

Start every classification with the nth-term test: if the terms do not tend to zero, the series diverges and no further work is allowed. If the terms do tend to zero, nothing is settled — the next step is pattern matching against the two benchmark families. A constant ratio between terms means geometric, convergent exactly when ∣r∣<1|r| < 1 with sum a/(1−r)a/(1-r); terms of size 1/np1/n^p mean a p-series, convergent exactly when p>1p > 1.

For positive-term series that merely resemble a benchmark, the comparison tests make the resemblance rigorous: direct comparison bounds the terms by a convergent series (or above a divergent one), and limit comparison only asks that the ratio of terms approach a positive constant, which is usually easier to verify for rational-function terms. For series built from factorials and exponentials, the ratio test is the natural tool, and the root test suits terms that are visibly nth powers; both deliver absolute convergence when their limit is below 1 and are silent at exactly 1 — the p-series family is the standard reminder, since every pp gives ratio limit 1.

Sign-alternating series get a two-stage analysis. First test the absolute-value series: if it converges, the original converges absolutely and behaves like a positive series. If it diverges, the alternating series test can still give convergence — decreasing magnitudes with limit zero suffice — and the result is conditional convergence, a genuinely more fragile state in which rearranging terms can change the sum. GRE questions probe exactly this distinction, so report which of the two kinds of convergence you have, not just that the series converges.

After this page, you should be able to

  • Classify geometric and p-series on sight and use them as comparison benchmarks.
  • Choose between the direct comparison and limit comparison tests for positive-term series.
  • Apply the ratio and root tests to series with factorials or nth powers, and state when each is inconclusive.
  • Distinguish absolute from conditional convergence and test each with the appropriate tool.

Formulas and assumptions

Geometric series sum

∑n=0∞arn=a1−r,∣r∣<1\sum_{n=0}^{\infty} a r^{n} = \frac{a}{1 - r}, \qquad |r| < 1

Variables

  • a: first term
  • r: common ratio

Assumptions

  • |r| < 1; for |r| >= 1 the series diverges.
  • Indexing from n = 0 with first term a.

p-series convergence

∑n=1∞1np converges  ⟺  p>1\sum_{n=1}^{\infty} \frac{1}{n^{p}} \text{ converges} \iff p > 1

Variables

  • p: fixed positive exponent
  • n: summation index

Assumptions

  • Established by the integral test.
  • p = 1 is the divergent harmonic series.

Ratio test

L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

Variables

  • a_n: series terms
  • L: limiting ratio

Assumptions

  • The limit L exists (or is infinite).
  • L = 1 gives no information; use another test.

Worked example

Classifying the alternating harmonic series with two tests

Determine whether ∑n=1∞(−1)n+1n\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} diverges, converges conditionally, or converges absolutely.

  1. 1Test absolute convergence first: the absolute-value series is ∑n=1∞1n\sum_{n=1}^{\infty} \dfrac{1}{n}, a p-series with p=1p = 1, which diverges. So the series does not converge absolutely.
  2. 2Apply the alternating series test to the original: the magnitudes bn=1nb_n = \dfrac{1}{n} are positive, strictly decreasing (1n+1<1n\tfrac{1}{n+1} < \tfrac{1}{n}), and lim⁡n→∞1n=0\lim_{n \to \infty} \dfrac{1}{n} = 0.
  3. 3Both hypotheses of the alternating series test hold, so the series converges.
  4. 4Combine the two results: convergent, but not absolutely convergent — the definition of conditional convergence.

The series converges conditionally: the alternating series test gives convergence, while the p-series test (p=1p = 1) shows the absolute-value series diverges.

Common traps

  • Concluding convergence because the terms tend to zero — the harmonic series fails exactly this way; the nth-term test can only prove divergence.
  • Reading anything into a ratio or root test limit of exactly 1; both tests are silent there and a p-series or comparison argument must take over.
  • Applying the alternating series test without checking that the magnitudes decrease — alternation with non-monotonic magnitudes can diverge.
  • Reporting "convergent" when a question distinguishes absolute from conditional convergence; the two-stage answer is the complete one.

Related pages and practice

GRE Mathematics overviewLimits and ContinuityEigenvalues and DeterminantsSign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

  1. GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
  2. Calculus Volume 2, Section 5.2: Infinite Series — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  3. Calculus Volume 2, Section 5.3: The Divergence and Integral Tests — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  4. Calculus Volume 2, Section 5.4: Comparison Tests — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  5. Calculus Volume 2, Section 5.5: Alternating Series — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  6. Calculus Volume 2, Section 5.6: Ratio and Root Tests — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.

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Recheck the ETS content-structure page and the OpenStax Calculus Volume 2 series-test sections before each major GRE Mathematics preparation cycle.

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