Math 2.0

Sequences, series, summability Theorem

Comparison test for improper integrals

Builds on Improper integralMonotonicity of the integralOne-sided limits of a monotone function exist

Statement

Let f,g ⁣:[a,∞)→ℝf, g \colon [a, \infty) \to \mathbb{R} be Riemann integrable on each [a,b][a, b], with 0≤f(x)≤g(x)0 \le f(x) \le g(x) for all x≥ax \ge a. If ∫a∞g(x) dx\int_a^\infty g(x) \, dx converges, then ∫a∞f(x) dx\int_a^\infty f(x) \, dx converges, and ∫a∞f(x) dx≤∫a∞g(x) dx.\int_a^\infty f(x) \, dx \le \int_a^\infty g(x) \, dx. If ∫a∞f(x) dx\int_a^\infty f(x) \, dx diverges, then ∫a∞g(x) dx\int_a^\infty g(x) \, dx diverges.

Remarks

The function b↦∫abfb \mapsto \int_a^b f is increasing, and it is bounded by ∫a∞g\int_a^\infty g by monotonicity. A bounded monotone function has a limit. For example, 0≤e−x2≤e−x0 \le e^{-x^2} \le e^{-x} for x≥1x \ge 1. So ∫1∞e−x2 dx\int_1^\infty e^{-x^2} \, dx converges. The same test holds for integrals that are improper at a finite endpoint.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementDefinite integral, definition, no evidence levelDefinite integraldefinitionImproper integral, definition, no evidence levelImproper integraldefinitionOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementComparison test for improper integrals, statement, no evidence levelComparison test for i…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.
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  1. Limit point of a set definition
  2. Partition of an interval definition
  3. Sequence definition
  4. Limit of a function definition
  5. Mesh of a partition definition
  6. Riemann sum definition
  7. Real sequence definition
  8. Limit at infinity definition
  9. One-sided limit definition
  10. Riemann integrable function definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Riemann integral definition
  15. Monotone convergence theorem for sequences statement
  16. Definite integral definition
  17. Improper integral definition
  18. One-sided limits of a monotone function exist statement
  19. Monotonicity of the integral statement
  20. Comparison test for improper integrals statement

Where this leads

32 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Convergence of the integral
About this page
Tag0FR1BDKK10
URIm2://0FR1BDKK10
Kindstatement, theorem
WritingW1 stated
MSC40A10
StatusThis copy of the site has no build data. Run m2 status 0FR1BDKK10.
UsesImproper integral, Monotonicity of the integral, One-sided limits of a monotone function exist

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