Math 2.0

Sequences, series, summability Concept

Sequence of functions

Builds on SequencePointwise convergenceUniform convergence

Summary

A sequence of functions is a sequence (fn)(f_n) of functions fn ⁣:S→ℝf_n \colon S \to \mathbb{R} on a common set SS. For each fixed x∈Sx \in S, the values fn(x)f_n(x) form a sequence of real numbers. Such sequences can converge in several ways, mainly pointwise or uniformly.

Remarks

The two modes give different properties of the limit. A uniform limit of continuous functions is continuous, but a pointwise limit need not be. Examples are the Taylor polynomials of a function, and the partial sums of a power series.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionSequence of functions, concept, no evidence levelSequence of functionsconcept
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. Limit of a sequence definition
  4. Pointwise convergence definition
  5. Uniform convergence definition
  6. Sequence of functions concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Function_series, Convergence
About this page
Tag0Y8FRYX8Q2
URIm2://0Y8FRYX8Q2
Kindconcept
WritingW0 stub
MSC40A30
SpecializesSequence
UsesSequence, Pointwise convergence, Uniform convergence

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