Math 2.0

Sequences, series, summability Theorem

Absolutely convergent series converge

Builds on Absolute convergenceCauchy criterion for series

Statement

Let (an)(a_n) be a real sequence. If the series ∑an\sum a_n converges absolutely, then ∑an\sum a_n converges. Moreover, ∣∑n=0∞an∣≤∑n=0∞∣an∣\left| \sum_{n=0}^{\infty} a_n \right| \le \sum_{n=0}^{\infty} |a_n|.

Remarks

The proof uses the Cauchy criterion for series. For n>mn > m the triangle inequality gives ∣∑k=m+1nak∣≤∑k=m+1n∣ak∣\left| \sum_{k=m+1}^{n} a_k \right| \le \sum_{k=m+1}^{n} |a_k|. The converse fails: the alternating harmonic series converges, but ∑1/n\sum 1/n diverges. The result holds in every complete normed space. Mathlib states it as Summable.of_norm.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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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…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionCauchy criterion for series, statement, no evidence levelCauchy criterion for …statementAbsolutely convergent series converge, statement, no evidence levelAbsolutely convergent…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. 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. Convergent series definition
  16. Cauchy criterion for sequences statement
  17. Absolute convergence definition
  18. Cauchy criterion for series statement
  19. Absolutely convergent series converge statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Relation to convergence
About this page
Tag0Y123X48GA
URIm2://0Y123X48GA
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0Y123X48GA.
UsesAbsolute convergence, Cauchy criterion for series

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