Convergent series in the context of "Analytic function"

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

In complex analysis, a complex-valued function ${\displaystyle f}$ of a complex variable ${\displaystyle z}$:

• is said to be holomorphic at a point ${\displaystyle a}$ if it is differentiable at every point within some open disk centered at ${\displaystyle a}$, and
• is said to be analytic at ${\displaystyle a}$ if in some open disk centered at ${\displaystyle a}$ it can be expanded as a convergent power series $${\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}$$ (this implies that the radius of convergence is positive).

One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. Among the corollaries of this theorem are

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series

A Fourier series (/ˈfʊrieɪ, -iər/) is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series § Definition.

The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a square wave.