Math 2.0

Sequences, series, summability Convergence tests Theorem

Ratio test

Builds on Absolute convergenceComparison testGeometric series

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 converges absolutely.

Remarks

Choose qq with L<q<1L < q < 1. For n≥Nn \ge N we get ∣an+1∣≤q∣an∣|a_{n+1}| \le q |a_n|, so ∣an∣≤∣aN∣qn−N|a_n| \le |a_N| q^{n-N}. The comparison test with a geometric series gives the result. For L>1L > 1 the series diverges, as the divergence part of the ratio test states. For L=1L = 1 the test gives no answer. The series ∑1/n\sum 1/n diverges and ∑1/n2\sum 1/n^2 converges, and both have L=1L = 1. For example, ∑xn/n!\sum x^n / n! converges for each real xx, since the ratio is ∣x∣/(n+1)→0|x|/(n+1) \to 0.

Reading path

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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 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 seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute 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 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.
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  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. Squeeze theorem statement
  12. Partial sum definition
  13. The sequence r^n tends to zero for |r| < 1 example
  14. Convergent series definition
  15. A series of nonnegative terms converges iff its partial sums are bounded statement
  16. Absolute convergence definition
  17. Sum of a series definition
  18. Comparison test statement
  19. Geometric series example
  20. Ratio test statement

Where this leads

73 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ratio_test, The test
About this page
Tag0QNY6ZBXPN
URIm2://0QNY6ZBXPN
Aliasesd-alembert-ratio-test
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0QNY6ZBXPN.
UsesAbsolute convergence, Comparison test, Geometric series
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