Editing Series (mathematics) (section)

==Series of functions==
{{Main|Function series}}
A series of real- or complex-valued functions

<math display=block>\sum_{n=0}^\infty f_n(x)</math>

is [[Pointwise convergence|pointwise convergent]] to a limit {{tmath|f(x)}} on a set {{tmath|E}} if the series converges for each {{tmath|x}} in {{tmath|E}} as a series of real or complex numbers. Equivalently, the partial sums

<math display=block>s_N(x) = \sum_{n=0}^N f_n(x)</math>

converge to {{tmath|f(x)}} as {{tmath|N}} goes to infinity for each {{tmath|x}} in {{tmath|E}}.

A stronger notion of convergence of a series of functions is [[uniform convergence]]. A series converges uniformly in a set <math>E</math> if it converges pointwise to the function {{tmath|f(x)}} at every point of <math>E</math> and the supremum of these pointwise errors in approximating the limit by the {{tmath|N}}th partial sum,

<math display=block>\sup_{x \in E} \bigl|s_N(x) - f(x)\bigr|</math>

converges to zero with increasing {{tmath|N}}, {{em|independently}} of {{tmath|x}}.

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 {{tmath|f_n}} are [[integral|integrable]] on a closed and bounded interval {{tmath|I}} and converge uniformly, then the series is also integrable on {{tmath|I}} and can be integrated term by term. Tests for uniform convergence include [[Weierstrass M-test|Weierstrass' M-test]], [[Abel's uniform convergence test]], [[Dini's test]], and the [[Cauchy sequence|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 set of [[null set|measure zero]]. Other [[modes of convergence]] depend on a different [[metric space]] structure on the [[Function space|space of functions]] under consideration. For instance, a series of functions '''converges in mean''' to a limit function {{tmath|f}} on a set {{tmath|E}} if

<math display=block>\lim_{N \rightarrow \infty} \int_E \bigl|s_N(x)-f(x)\bigr|^2\,dx = 0.</math>

===Power series===
:{{Main|Power series}}

A '''power series''' is a series of the form

<math display=block>\sum_{n=0}^\infty a_n(x-c)^n.</math>

The [[Taylor series]] at a point {{tmath|c}} of a function is a power series that, in many cases, converges to the function in a neighborhood of {{tmath|c}}. For example, the series

<math display=block>\sum_{n=0}^{\infty} \frac{x^n}{n!}</math>

is the Taylor series of <math>e^x</math> at the origin and converges to it for every {{tmath|x}}.

Unless it converges only at {{tmath|1= x = c}}, such a series converges on a certain open disc of convergence centered at the point {{tmath|c}} in the complex plane, and may also converge at some of the points of the boundary of the disc.  The radius of this disc is known as the [[radius of convergence]], and can in principle be determined from the asymptotics of the coefficients {{tmath|a_n}}.  The convergence is uniform on [[closed set|closed]] and [[bounded set|bounded]] (that is, [[compact set|compact]]) subsets of the interior of the disc of convergence: to wit, it is [[Compact convergence|uniformly convergent on compact sets]].

Historically, mathematicians such as [[Leonhard Euler]] operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.

=== Formal power series ===
{{main|Formal power series}}

While many uses of power series refer to their sums, it is also possible to treat power series as ''formal sums'', meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in [[combinatorics]] to describe and study [[sequence]]s that are otherwise difficult to handle, for example, using the method of [[generating function]]s.  The [[Hilbert–Poincaré series]] is a formal power series used to study [[graded algebra]]s.

Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as [[addition]], [[multiplication]], [[derivative]], [[antiderivative]] for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a [[commutative ring]], so that the formal power series can be added term-by-term and multiplied via the [[Cauchy product]]. In this case the algebra of formal power series is the [[total algebra]] of the [[monoid]] of [[natural numbers]] over the underlying term ring.<ref>{{citation|author=Nicolas Bourbaki|author-link=Nicolas Bourbaki|title=Algebra|publisher=Springer|year=1989}}: §III.2.11.</ref>  If the underlying term ring is a [[differential algebra]], then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.

===Laurent series===
{{Main|Laurent series}}
Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents.  A Laurent series is thus any series of the form

<math display=block>\sum_{n=-\infty}^\infty a_n x^n.</math>

If such a series converges, then in general it does so in an [[annulus (mathematics)|annulus]] rather than a disc, and possibly some boundary points.  The series converges uniformly on compact subsets of the interior of the annulus of convergence.

===Dirichlet series===
:{{Main|Dirichlet series}}

A [[Dirichlet series]] is one of the form

<math display=block>\sum_{n=1}^\infty {a_n \over n^s},</math>

where {{tmath|s}} is a [[complex number]].  For example, if all {{tmath|a_n}} are equal to {{tmath|1}}, then the Dirichlet series is the [[Riemann zeta function]]

<math display=block>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>

Like the zeta function, Dirichlet series in general play an important role in [[analytic number theory]].  Generally a Dirichlet series converges if the real part of {{tmath|s}} is greater than a number called the abscissa of convergence.  In many cases, a Dirichlet series can be extended to an [[analytic function]] outside the domain of convergence by [[analytic continuation]].  For example, the Dirichlet series for the zeta function converges absolutely when {{tmath|\operatorname{Re}(s)>1}}, but the zeta function can be extended to a holomorphic function defined on <math>\Complex\setminus\{1\}</math> with a simple [[pole (complex analysis)|pole]] at&nbsp;{{tmath|1}}.

This series can be directly generalized to [[general Dirichlet series]].

===Trigonometric series===
{{Main|Trigonometric series}}
A series of functions in which the terms are [[trigonometric function]]s is called a '''trigonometric series''':

<math display=block>A_0 + \sum_{n=1}^\infty \left(A_n\cos nx + B_n \sin nx\right).</math>

The most important example of a trigonometric series is the [[Fourier series]] of a function.

=== Asymptotic series ===
{{Main|Asymptotic expansion}}
[[Asymptotic series]], typically called [[asymptotic expansion]]s, are infinite series whose terms are functions of a sequence of different [[Big O notation|asymptotic orders]] and whose partial sums are approximations of some other function in an [[asymptotic limit]]. In general they do not converge, but they are still useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. They are crucial tools in [[perturbation theory]] and in the [[analysis of algorithms]].

An asymptotic series cannot necessarily be made to produce an answer as exactly as desired away from the asymptotic limit, the way that an ordinary convergent series of functions can. In fact, a typical asymptotic series reaches its best practical approximation away from the asymptotic limit after a finite number of terms; if more terms are included, the series will produce less accurate approximations.