Math 2.0

Sequences, series, summability Convergence tests Theorem

Root test

Builds on Limit superiorAbsolute convergenceComparison testGeometric series

Statement

Let (an)(a_n) be a real sequence and let C=lim sup⁡n→∞∣an∣1/nC = \limsup_{n \to \infty} |a_n|^{1/n}, the limit superior, with the value +∞+\infty allowed. If C<1C < 1, then the series ∑an\sum a_n converges absolutely. If C>1C > 1, then the series ∑an\sum a_n diverges.

Remarks

If C<1C < 1, choose qq with C<q<1C < q < 1. Then ∣an∣≤qn|a_n| \le q^n for large nn, and the comparison test with a geometric series applies. If C>1C > 1, then ∣an∣>1|a_n| > 1 for infinitely many nn, so the terms do not tend to 00. For C=1C = 1 the test gives no answer. The root test is stronger than the ratio test: when the ratio limit exists, the root limit exists and has the same value.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionLimit superior, definition, no evidence levelLimit superiordefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRoot test, statement, no evidence levelRoot teststatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Limit superior definition
  9. Sum of a finite geometric series statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Squeeze theorem statement
  13. Partial sum definition
  14. The sequence r^n tends to zero for |r| < 1 example
  15. Convergent series definition
  16. Absolute convergence definition
  17. A series of nonnegative terms converges iff its partial sums are bounded statement
  18. Sum of a series definition
  19. Comparison test statement
  20. Geometric series example
  21. Root test statement

Where this leads

29 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Root_test, Root test explanation
About this page
Tag05H4DDDYBZ
URIm2://05H4DDDYBZ
Aliasescauchy-root-test
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 05H4DDDYBZ.
UsesLimit superior, Absolute convergence, Comparison test, Geometric series
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