Math 2.0

Sequences, series, summability Proposition

Divergence part of the ratio test

Builds on Ratio testTerm test

Statement

Let (an)(a_n) be a real sequence with an≠0a_n \ne 0 for all nn. Assume that ∣an+1∣/∣an∣→L|a_{n+1}| / |a_n| \to L as n→∞n \to \infty, for a real number LL with L>1L > 1. Then the series ∑an\sum a_n diverges.

Remarks

For large nn we have ∣an+1∣>∣an∣|a_{n+1}| > |a_n|. So ∣an∣|a_n| is eventually increasing and positive, and ana_n does not tend to 00. The term test gives divergence. The same proof works when ∣an+1∣/∣an∣→+∞|a_{n+1}| / |a_n| \to +\infty. For example, ∑2n/n\sum 2^n / n diverges, since the ratio is 2n/(n+1)→22n/(n+1) \to 2.

Reading path

To understand this page from the ground up, read these 24 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 levelSeriesdefinitionSum 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 sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSqueeze 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…exampleLimit of a sum of sequences, statement, no evidence levelLimit of a sum of seq…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementTerm test, statement, no evidence levelTerm teststatementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementDivergence part of the ratio test, statement, no evidence levelDivergence part of th…statement
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. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Epsilon-N definition of the limit of a sequence statement
  12. Squeeze theorem statement
  13. Partial sum definition
  14. The sequence r^n tends to zero for |r| < 1 example
  15. Limit of a sum of sequences statement
  16. Convergent series definition
  17. A series of nonnegative terms converges iff its partial sums are bounded statement
  18. Term test statement
  19. Absolute convergence definition
  20. Sum of a series definition
  21. Comparison test statement
  22. Geometric series example
  23. Ratio test statement
  24. Divergence part of the ratio test statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Ratio_test, The test
About this page
Tag08AM0KRCZ1
URIm2://08AM0KRCZ1
Kindstatement, proposition
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 08AM0KRCZ1.
UsesRatio test, Term test

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