Math 2.0

statement theorem

Term-by-term integration

Statement

Let a<ba < b, and let fn ⁣:[a,b]→Rf_n \colon [a, b] \to \mathbb{R} be Riemann integrable for each n≥0n \ge 0. If ∑nfn\sum_{n} f_n converges uniformly on [a,b][a, b], then its sum is Riemann integrable, and ∫ab∑n=0∞fn(x) dx=∑n=0∞∫abfn(x) dx.\int_a^b \sum_{n=0}^{\infty} f_n(x) \, dx = \sum_{n=0}^{\infty} \int_a^b f_n(x) \, dx.

Remarks

Apply the theorem on sequences to the partial sums. For example, integrating 11+t=∑(−t)n\frac{1}{1 + t} = \sum (-t)^n term by term over [0,x][0, x], with ∣x∣<1|x| < 1, gives the Mercator series for ln⁡(1+x)\ln(1 + x). Mathlib states a version with summable sup norms as intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm.

Explanations

L1 human

We may integrate a uniformly convergent series term by term. The integral of the sum is the sum of the integrals.

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To read this node, read these 21 nodes in this order. The foundations come first, and the list ends with this node.

Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionUniform convergence, definition, no evidence levelUniform convergencedefinitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionRiemann's criterion for integrability, statement, no evidence levelRiemann's criterion f…statementUniform limits commute with integration, statement, no evidence levelUniform limits commut…statementTerm-by-term integration, statement, no evidence levelTerm-by-term integrat…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Partition of an interval definition
  2. Sequence definition
  3. Mesh of a partition definition
  4. Riemann sum definition
  5. Real sequence definition
  6. Riemann integrable function definition
  7. Bounded sequence definition
  8. Limit of a sequence definition
  9. Series definition
  10. Riemann integral definition
  11. Upper and lower Darboux sums definition
  12. Pointwise convergence definition
  13. Definite integral definition
  14. Upper and lower Darboux integrals definition
  15. Uniform convergence definition
  16. Darboux and Riemann integrals agree statement
  17. Monotonicity of the integral statement
  18. Series of functions definition
  19. Riemann's criterion for integrability statement
  20. Uniform limits commute with integration statement
  21. Term-by-term integration statement

Sources

  1. Web https://en.wikipedia.org/wiki/Function_series, Convergence

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