Math 2.0

concept

Sequence of functions

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.

Explanations

L1 human

A sequence of functions is a list of graphs, one for each nn. We ask whether the graphs approach the graph of a limit function, and in what sense.

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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 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.
  • 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

Sources

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

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