Math 2.0

Sequences, series, summability Definition

Absolutely convergent improper integral

Builds on Improper integral

Definition

Let f ⁣:[a,∞)→ℝf \colon [a, \infty) \to \mathbb{R} be Riemann integrable on [a,b][a, b] for each b>ab > a. The improper integral ∫a∞f(x) dx\int_a^\infty f(x) \, dx converges absolutely if ∫a∞∣f(x)∣ dx\int_a^\infty |f(x)| \, dx converges.

Remarks

An absolutely convergent improper integral converges. The proof uses 0≤f+∣f∣≤2∣f∣0 \le f + |f| \le 2|f| and the comparison test. The converse fails. The Dirichlet integral ∫0∞sin⁡xx dx\int_0^\infty \frac{\sin x}{x} \, dx converges, but not absolutely. A function on [a,∞)[a, \infty) whose improper integral converges absolutely is Lebesgue integrable there. The two integrals agree.

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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…definitionLimit 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 sumdefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionRiemann integral, definition, no evidence levelRiemann integraldefinitionImproper integral, definition, no evidence levelImproper integraldefinitionAbsolutely convergent improper integral, definition, no evidence levelAbsolutely convergent…definition
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  1. Limit point of a set definition
  2. Partition of an interval definition
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  5. Riemann sum definition
  6. Limit at infinity definition
  7. One-sided limit definition
  8. Riemann integrable function definition
  9. Riemann integral definition
  10. Improper integral definition
  11. Absolutely convergent improper integral definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Improper Riemann integrals and Lebesgue integrals
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