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Root Test

Classify an infinite series using the limit superior of the nth roots of its term magnitudes: below one gives absolute convergence, above one gives divergence, and one is inconclusive.

Version
v2 · 2026-09-06 · History
Domain-specific #
2693
Origin domain
mathematics
Subdomain
real analysis
Aliases
Cauchy root test, Cauchy's radical test

Core Idea

For a series \(\sum_{n=1}^{\infty}a_n\), define

\[ L=\limsup_{n\to\infty}|a_n|^{1/n}. \]

The root test proves absolute convergence when \(L<1\) and divergence when \(L>1\), including \(L=+\infty\). When \(L=1\), it gives no conclusion. The proof compares eventual term magnitudes with a geometric sequence.

The recognition invariant is nth-root exponential scale + limsup + threshold at one + three-way conclusion.

Scope of Application

The test is effective when terms contain nth powers, factorial-like exponential scales, or irregular successive ratios. Applied to \(\sum c_n(z-z_0)^n\), it yields the Cauchy–Hadamard radius \(R^{-1}=\limsup |c_n|^{1/n}\), with extended-value conventions.

It classifies absolute convergence. Conditional convergence requires another method when the absolute-value series lands at the boundary.

Clarity

If the ordinary limit of \(|a_n|^{1/n}\) exists, it equals the limsup and may be used. Otherwise, replacing limsup by a nonexistent limit is invalid. Above one, infinitely many terms fail to approach zero rapidly enough; the theorem need not claim they grow monotonically.

Manages Complexity

Nth roots discard subexponential factors and expose long-run exponential scale. Complicated products and powers often reduce to a single threshold comparison. The limsup makes the criterion robust to bounded oscillation and exceptional subsequences.

Abstract Reasoning

  1. Form \(|a_n|^{1/n}\).
  2. Compute or bound its limsup.
  3. If below one, choose \(q\) strictly between \(L\) and one and compare the tail with \(q^n\).
  4. If above one, find infinitely many terms whose magnitudes do not tend to zero.
  5. If equal to one, stop and select another test.
  6. For power series, multiply the coefficient rate by \(|z-z_0|\).

Knowledge Transfer

The portable structure is converting multiplicative growth into an asymptotic rate and comparing that rate with a stability threshold. The proposed immediate parent is Convergence.

Relationships to Other Abstractions

Local relationship map for Root TestParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Root TestDOMAINPrime abstraction: Convergence — is a kind ofConvergencePRIME

Current abstraction Root Test Domain-specific

Parents (1) — more general patterns this builds on

  • Root Test is a kind of Convergence Prime

    Convergence is the proposed immediate parent.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Root Test sits in a sparse region of the domain-specific corpus (94th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Special Functions & Convergence Tests (6 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08