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Infinite Series

Root Test

Definition

Root Test

Let ∑n=1∞an\sum_{n=1}^{\infty} a_n be a positive series with an≥0a_n \geq 0. If:

lim⁡n→∞ann=ρ\lim_{n \to \infty} \sqrt[n]{a_n} = \rho

then:

  1. When ρ<1\rho < 1, the series converges
  2. When ρ>1\rho > 1, the series diverges
  3. When ρ=1\rho = 1, the test is inconclusive
符号说明
SymbolTypePronunciation/ExplanationMeaning in This Article
ρ\rhoGreek letterRhoRepresents the limit value in series convergence tests
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
lim⁡\limMathematical symbolLimitRepresents limit of sequence or function

Formula

Root Test Formula

lim⁡n→∞ann=ρ\lim_{n \to \infty} \sqrt[n]{a_n} = \rho

  • When ρ<1\rho < 1, the series converges
  • When ρ>1\rho > 1, the series diverges
  • When ρ=1\rho = 1, the test is inconclusive

Applicable Situations

  • When the general term contains powers of nn
  • When the ratio test fails

Summary

Symbols Appearing in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
ρ\rhoGreek letterRho (row)Represents the limit value in series convergence tests
∑\sumGreek letterSigma (sigma)Summation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
lim⁡\limMathematical symbolLimitRepresents the limit of a sequence or function
ann\sqrt[n]{a_n}Mathematical symboln-th rootThe n-th root of ana_n

Chinese-English Glossary

Chinese TermEnglish TermPhoneticExplanation
Root testroot test/ruːt test/Method to determine convergence using n-th root
Cauchy’s testCauchy’s test/ˈkoʊʃiz test/Another name for the root test
Positive seriespositive series/ˈpɒzətɪv ˈsɪəriːz/Series where all terms are non-negative
Convergenceconvergence/kənˈvɜːdʒəns/Series partial sums have a finite limit
Divergencedivergence/daɪˈvɜːdʒəns/Series partial sums have no finite limit
PreviousRatio TestLearn the definition, usage, and applications of the ratio test (D'Alembert's test). NextIntegral TestLearn the definition, usage, and applications of the integral test.

课程路线图

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    Sequences bridge discrete thinking and calculus. This track covers core definitions, limits, convergence, and classic models.

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    The World of Limits in Advanced Mathematics

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    Explore convergence tests, summation, power-series expansions, and applications.

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