Math 2.0

Sequences, series, summability Series: basic notions Theorem

Cauchy criterion for series

Builds on Cauchy criterion for sequencesPartial sum

Statement

Let (an)(a_n) be a real sequence. The series ∑an\sum a_n converges if and only if the following holds for each ε>0\varepsilon > 0. There is a natural number NN such that ∣∑k=m+1nak∣<εfor all natural numbers n>m≥N.\left| \sum_{k=m+1}^{n} a_k \right| < \varepsilon \quad \text{for all natural numbers } n > m \ge N.

Remarks

The sum in the condition is sn−sms_n - s_m, a difference of partial sums. So the condition says that (sn)(s_n) is a Cauchy sequence, and the result is the Cauchy criterion for sequences. With n=m+1n = m + 1 it gives the term test. It proves that absolutely convergent series converge.

Reading path

To understand this page from the ground up, read these 16 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…statementCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementCauchy criterion for series, statement, no evidence levelCauchy criterion for …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. Cauchy criterion for sequences statement
  16. Cauchy criterion for series statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_convergence_test, Statement
About this page
Tag06PVXMY0TH
URIm2://06PVXMY0TH
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 06PVXMY0TH.
UsesCauchy criterion for sequences, Partial sum
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