section_path: Series of functions
section_name: Series of functions
paragraph_text: Series of functions
  
  A series of real- or complex-valued functions
  
  ∑ n = 0 ∞ f n (x) {\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}
  
  converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers. Equivalently, the partial sums
  
  s N (x) = ∑ n = 0 N f n (x) {\displaystyle s_{N}(x)=\sum _{n=0}^{N}f_{n}(x)}
  
  converge to ƒ (x) as N → ∞ for each x ∈ E.
  
  A stronger notion of convergence of a series of functions is the uniform convergence. A series converges uniformly if it converges pointwise to the function ƒ (x), and the error in approximating the limit by the N th partial sum,
  
  | s N (x) − f (x) | {\displaystyle |s_{N}(x)-f(x)|}
  
  can be made minimal independently of x by choosing a sufficiently large N.
  
  Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ƒ are integrable on a closed and bounded interval I and converge uniformly, then the series is also integrable on I and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.
  
  More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a certain set of measure zero. Other modes of convergence depend on a different metric space structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set E to a limit function ƒ provided
  
  ∫ E | s N (x) − f (x) | 2 d x → 0 {\displaystyle \int _{E}\left|s_{N}(x)-f(x)\right|^{2}\,dx\to 0}
  
  as N → ∞.