Math 2.0

statement theorem

Absolutely convergent series converge

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.

Explanations

L1 human

Cancellation between terms of opposite signs can only make partial sums smaller in absolute value. So if the series converges without any cancellation, it also converges with it.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Relation to convergence

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