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A series of real-valued functions and its pointwise and uniform convergence through its partial sums
Definition
Let be a set and let for . The series of real-valued functions is studied through its partial-sum functions
where the sum on the right is the finite sum of Series, partial sums, convergence and the sum, divergence, and the tail series. Thus is the zero function and under the pointwise operations of The vector space of all functions with pointwise operations, and as the case .
The series converges pointwise to when pointwise, and it converges uniformly to when uniformly (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).
The series is absolutely convergent at when the scalar series converges. It is absolutely pointwise convergent when this holds for every .
Depends on
- The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$
- Series, partial sums, convergence and the sum, divergence, and the tail series
- Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions
Used by
- A series of real-valued functions converges uniformly if and only if its tails are uniformly small Corollary
- The classical Weierstrass function Definition
- The tent function φ(t)=dist(t,ℤ) and the Takagi series T(x)=∑_n≥02⁻ⁿφ(2ⁿ x) Definition
- A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence Theorem
- The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series Theorem
Dependency tree · two levels
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Sources
- J. Lebl, Basic Analysis I, §6.1 (standard reference, not scraped)
- W. Trench, Introduction to Real Analysis (standard reference, not scraped)