Math 2.0

Sequences, series, summability Convergence tests Theorem

Integral test

Builds on A series of nonnegative terms converges iff its partial sums are boundedImproper integralComparison of a series and an integral

Statement

Let NN be an integer and let f ⁣:[N,∞)→ℝf \colon [N, \infty) \to \mathbb{R} be a decreasing function with f(x)≥0f(x) \ge 0 for all x≥Nx \ge N. Then the series ∑n=N∞f(n)\sum_{n=N}^{\infty} f(n) converges if and only if the improper integral ∫N∞f(x) dx\int_N^{\infty} f(x)\, dx converges.

Remarks

A decreasing function is Riemann integrable on each bounded interval, so the integrals ∫Nbf(x) dx\int_N^b f(x)\, dx exist. The proof uses the comparison of a series and an integral. When both converge, ∫N∞f(x) dx≤∑n=N∞f(n)≤f(N)+∫N∞f(x) dx.\int_N^{\infty} f(x)\, dx \le \sum_{n=N}^{\infty} f(n) \le f(N) + \int_N^{\infty} f(x)\, dx. With f(x)=1/xpf(x) = 1/x^p it gives the p-series test. Mathlib states the convergence direction for an antitone function, in AntitoneOn.summable_of_integrableOn_Ioi.

Reading path

To understand this page from the ground up, read these 23 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 sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPartial sum, definition, no evidence levelPartial sumdefinitionComparison of a series and an integral, statement, no evidence levelComparison of a serie…statementImproper integral, definition, no evidence levelImproper integraldefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementIntegral test, statement, no evidence levelIntegral teststatement
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.
  • no evidence level
  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. Series definition
  15. Riemann integral definition
  16. Convergent sequence definition
  17. Monotone convergence theorem for sequences statement
  18. Partial sum definition
  19. Comparison of a series and an integral statement
  20. Improper integral definition
  21. Convergent series definition
  22. A series of nonnegative terms converges iff its partial sums are bounded statement
  23. Integral test statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral_test_for_convergence, Statement of the test
About this page
Tag019QZ6RRVH
URIm2://019QZ6RRVH
Aliasesintegral-test-for-convergence
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 019QZ6RRVH.
UsesA series of nonnegative terms converges iff its partial sums are bounded, Improper integral, Comparison of a series and an integral
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