Math 2.0

Sequences, series, summability Convergence tests Theorem

Cauchy condensation test

Builds on A series of nonnegative terms converges iff its partial sums are boundedMonotone sequence

Statement

Let (an)n≥1(a_n)_{n \ge 1} be a decreasing real sequence with an≥0a_n \ge 0 for all nn. Then ∑n=1∞an\sum_{n=1}^{\infty} a_n converges if and only if the condensed series ∑k=0∞2ka2k\sum_{k=0}^{\infty} 2^k a_{2^k} converges.

Remarks

The proof groups the terms from 2k2^k to 2k+1−12^{k+1} - 1. Since the sequence decreases, ∑n=1∞an≤∑k=0∞2ka2k≤2∑n=1∞an\sum_{n=1}^{\infty} a_n \le \sum_{k=0}^{\infty} 2^k a_{2^k} \le 2 \sum_{n=1}^{\infty} a_n. With an=1/npa_n = 1/n^p it gives the p-series test. With an=1/(nln⁡n)a_n = 1/(n \ln n) for n≥2n \ge 2, the condensed series is a multiple of ∑1/k\sum 1/k, so ∑1/(nln⁡n)\sum 1/(n \ln n) diverges.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementCauchy condensation test, statement, no evidence levelCauchy condensation t…statement
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Partial sum definition
  10. Convergent series definition
  11. A series of nonnegative terms converges iff its partial sums are bounded statement
  12. Cauchy condensation test statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_condensation_test, Estimate
About this page
Tag0B96KMDDD1
URIm2://0B96KMDDD1
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0B96KMDDD1.
UsesA series of nonnegative terms converges iff its partial sums are bounded, Monotone sequence
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