Math 2.0

Sequences, series, summability Convergence tests Theorem

Dirichlet's test

Builds on Monotone sequencePartial sumCauchy criterion for series

Statement

Let (an)(a_n) be a monotone real sequence with an→0a_n \to 0. Let (bn)(b_n) be a real sequence whose partial sums are bounded. That is, there is MM with ∣∑k=1nbk∣≤M\left| \sum_{k=1}^{n} b_k \right| \le M for all nn. Then the series ∑n=1∞anbn\sum_{n=1}^{\infty} a_n b_n converges.

Remarks

The proof uses summation by parts and the Cauchy criterion for series. With bn=(−1)nb_n = (-1)^n it gives the alternating series test. It also shows that ∑sin⁡(nx)/n\sum \sin(n x) / n converges for each real xx. Mathlib states a form with a Cauchy sequence of partial sums, as Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded.

History

The test is named after Peter Gustav Lejeune Dirichlet. It was published after his death, in 1862.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPartial sum, definition, no evidence levelPartial sumdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementCauchy criterion for series, statement, no evidence levelCauchy criterion for …statementDirichlet's test, statement, no evidence levelDirichlet's 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. Subsequence definition
  4. Bounded sequence definition
  5. Cauchy sequence definition
  6. Limit of a sequence definition
  7. Monotone sequence definition
  8. Series definition
  9. Cauchy sequences are bounded statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Every sequence has a monotone subsequence statement
  13. Partial sum definition
  14. Bolzano–Weierstrass theorem statement
  15. Cauchy criterion for sequences statement
  16. Cauchy criterion for series statement
  17. Dirichlet's test statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet%27s_test, Statement
About this page
Tag0V9ME8GQE7
URIm2://0V9ME8GQE7
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0V9ME8GQE7.
UsesMonotone sequence, Partial sum, Cauchy criterion for series
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