Math 2.0

Sequences, series, summability Convergence tests Theorem

Comparison test

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

Statement

Let (an)(a_n) and (bn)(b_n) be real sequences with 0≤an≤bn0 \le a_n \le b_n for all nn. If ∑bn\sum b_n converges, then ∑an\sum a_n converges, and ∑an≤∑bn\sum a_n \le \sum b_n. If ∑an\sum a_n diverges, then ∑bn\sum b_n diverges.

Remarks

The partial sums of ∑an\sum a_n are at most those of ∑bn\sum b_n. The result follows from a series of nonnegative terms converges iff its partial sums are bounded. It is enough that 0≤an≤bn0 \le a_n \le b_n for all nn larger than some NN. For example, 0≤1/(n2+1)≤1/n20 \le 1/(n^2 + 1) \le 1/n^2, so ∑1/(n2+1)\sum 1/(n^2 + 1) converges. Mathlib states it for a summable upper bound, in Summable.of_nonneg_of_le.

Reading path

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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…statementComparison test, statement, no evidence levelComparison teststatement
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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. Comparison test statement

Where this leads

118 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_comparison_test, For series
About this page
Tag04Z5G0THN6
URIm2://04Z5G0THN6
Aliasesdirect-comparison-test
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 04Z5G0THN6.
UsesA series of nonnegative terms converges iff its partial sums are bounded
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