Math 2.0

definition

Absolutely convergent 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.

Explanations

L1 human

An improper integral converges absolutely when the area between the graph and the axis is finite, with all parts counted as positive. This is the analogue of absolute convergence of series.

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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
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  8. Riemann integrable function definition
  9. Riemann integral definition
  10. Improper integral definition
  11. Absolutely convergent improper integral definition

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Sources

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

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