Math 2.0

Sequences, series, summability Absolute and conditional convergence, rearrangements, products of series Theorem

Weierstrass M-test

Builds on Series of functionsComparison testUniformly Cauchy sequence of functions

Statement

Let SS be a set and let fn ⁣:S→ℝf_n \colon S \to \mathbb{R} for n≥0n \ge 0. Assume that there are real numbers Mn≥0M_n \ge 0 with ∣fn(x)∣≤Mn|f_n(x)| \le M_n for all nn and all x∈Sx \in S, and ∑n=0∞Mn<∞\sum_{n=0}^{\infty} M_n < \infty. Then the series of functions ∑n=0∞fn\sum_{n=0}^{\infty} f_n converges absolutely and uniformly on SS.

Remarks

Absolute convergence at each xx follows from the comparison test. For m<nm < n, ∣sn(x)−sm(x)∣≤∑k=m+1nMk|s_n(x) - s_m(x)| \le \sum_{k=m+1}^{n} M_k. So the partial sums are uniformly Cauchy. For example, ∑sin⁡(nx)n2\sum \frac{\sin(n x)}{n^2} converges uniformly on ℝ\mathbb{R}, with Mn=1/n2M_n = 1/n^2. Mathlib states it as tendstoUniformlyOn_tsum.

Reading path

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionComparison test, statement, no evidence levelComparison teststatementWeierstrass M-test, statement, no evidence levelWeierstrass M-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. Sequence definition
  2. Real sequence definition
  3. Subsequence definition
  4. Bounded sequence definition
  5. Cauchy sequence definition
  6. Limit of a sequence definition
  7. Monotone sequence definition
  8. Series definition
  9. Cauchy sequences are bounded statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Every sequence has a monotone subsequence statement
  13. Pointwise convergence definition
  14. Partial sum definition
  15. Bolzano–Weierstrass theorem statement
  16. Convergent series definition
  17. Uniform convergence definition
  18. Cauchy criterion for sequences statement
  19. Uniform convergence on a set definition
  20. A series of nonnegative terms converges iff its partial sums are bounded statement
  21. Series of functions definition
  22. Uniformly Cauchy sequence of functions definition
  23. Comparison test statement
  24. Weierstrass M-test statement

Where this leads

43 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Weierstrass_M-test, Statement
About this page
Tag0NX5NWAFC9
URIm2://0NX5NWAFC9
Kindstatement, theorem
WritingW1 stated
MSC40A30
StatusThis copy of the site has no build data. Run m2 status 0NX5NWAFC9.
UsesSeries of functions, Comparison test, Uniformly Cauchy sequence of functions
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