Math 2.0

Sequences, series, summability Theorem

Term-by-term differentiation

Builds on Uniform convergence of derivatives allows term-by-term differentiationSeries of functions

Statement

Let a<ba < b, and let fn ⁣:[a,b]→ℝf_n \colon [a, b] \to \mathbb{R} be differentiable for each n≥0n \ge 0. Assume that ∑nfn(x0)\sum_{n} f_n(x_0) converges for some x0∈[a,b]x_0 \in [a, b], and that ∑nfn′\sum_{n} f_n' converges uniformly on [a,b][a, b]. Then ∑nfn\sum_{n} f_n converges uniformly on [a,b][a, b], its sum is differentiable, and (∑n=0∞fn)′=∑n=0∞fn′.\Big( \sum_{n=0}^{\infty} f_n \Big)' = \sum_{n=0}^{\infty} f_n'.

Remarks

Apply the theorem on sequences to the partial sums. The derivative series often converges uniformly by the M-test. Mathlib states that form as hasDerivAt_tsum. Power series satisfy the hypotheses on each closed interval inside the interval of convergence.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded 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 levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementCauchy 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 convergencedefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementUniform convergence of derivatives allows term-by-term differentiation, statement, no evidence levelUniform convergence o…statementTerm-by-term differentiation, statement, no evidence levelTerm-by-term differen…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.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Series definition
  15. Continuous function definition
  16. Fermat's theorem on stationary points statement
  17. Cauchy sequences are bounded statement
  18. Monotone convergence theorem for sequences statement
  19. Convergent sequence definition
  20. Every sequence has a monotone subsequence statement
  21. Pointwise convergence definition
  22. Bolzano–Weierstrass theorem statement
  23. Uniform convergence definition
  24. Boundedness theorem statement
  25. Uniform convergence on a set definition
  26. Cauchy criterion for sequences statement
  27. Series of functions definition
  28. Extreme value theorem statement
  29. Uniformly Cauchy sequence of functions definition
  30. Rolle's theorem statement
  31. Mean value theorem statement
  32. Uniform convergence of derivatives allows term-by-term differentiation statement
  33. Term-by-term differentiation statement

Where this leads

28 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Function_series, Convergence
About this page
Tag0TJ8QBJX5K
URIm2://0TJ8QBJX5K
Kindstatement, theorem
WritingW0 stub
MSC40A30, 26A24
StatusThis copy of the site has no build data. Run m2 status 0TJ8QBJX5K.
UsesUniform convergence of derivatives allows term-by-term differentiation, Series of functions

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