Editing Real analysis (section)

===Series===
{{Main|Series (mathematics)}}
A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.  The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.  The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms.  Instead, the finite sum of the first <math>n</math> terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as <math>n</math> grows without bound.  The series is assigned the value of this limit, if it exists.

Given an (infinite) [[sequence]] <math>(a_n)</math>, we can define an associated '''''series''''' as the formal mathematical object {{nowrap|<math display="inline">a_1 + a_2 + a_3 + \cdots = \sum_{n=1}^{\infty} a_n</math>,}} sometimes simply written as <math display="inline">\sum a_n</math>.  The '''''partial sums''''' of a series <math display="inline">\sum a_n</math> are the numbers <math display="inline">s_n=\sum_{j=1}^n a_j</math>.  A series <math display="inline">\sum a_n</math> is said to be '''''convergent''''' if the sequence consisting of its partial sums, <math>(s_n)</math>, is convergent; otherwise it is '''''divergent'''''.  The '''''sum''''' of a convergent series is defined as the number {{nowrap|<math display="inline">s = \lim_{n \to \infty} s_n</math>.}}

The word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms.  For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the ''[[Riemann rearrangement theorem]]'' for further discussion).

An example of a convergent series is a [[geometric series]] which forms the basis of one of Zeno's famous [[Zeno's paradoxes|paradoxes]]:

:<math>\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1 .</math>

In contrast, the [[Harmonic series (mathematics)|harmonic series]] has been known since the Middle Ages to be a divergent series:

:<math>\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty .</math>

(Here, "<math>=\infty</math>" is merely a notational convention to indicate that the partial sums of the series grow without bound.)

A series <math display="inline">\sum a_n</math> is said to '''''[[Absolute convergence|converge absolutely]]''''' if <math display="inline">\sum |a_n|</math> is convergent.  A convergent series <math display="inline">\sum a_n</math> for which <math display="inline">\sum |a_n|</math> diverges is said to '''''converge''''' '''''non-absolutely'''''.<ref>The term '''''unconditional convergence''''' refers to series whose sum does not depend on the order of the terms (i.e., any rearrangement gives the same sum).  Convergence is termed '''''conditional''''' otherwise.  For series in <math>\R^n</math>, it can be shown that absolute convergence and unconditional convergence are equivalent.  Hence, the term "conditional convergence" is often used to mean non-absolute convergence.  However, in the general setting of Banach spaces, the terms do not coincide, and there are unconditionally convergent series that do not converge absolutely.</ref>  It is easily shown that absolute convergence of a series implies its convergence.  On the other hand, an example of a series that converges non-absolutely is

:<math>\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2 .</math>

====Taylor series====
{{Main|Taylor series}}
The Taylor series of a [[real-valued function|real]] or [[complex-valued function]] ''ƒ''(''x'') that is [[infinitely differentiable function|infinitely differentiable]] at a [[real number|real]] or [[complex number]] ''a'' is the [[power series]]
<!--
As stated below, the Taylor series need not equal the function. So please don't write f(x)=... here. In other words,

DO NOT CHANGE ANYTHING ABOUT THIS FORMULA-->:<math>f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f^{(3)}(a)}{3!} (x-a)^3 + \cdots. </math><!---->

which can be written in the more compact [[Summation#Capital-sigma notation|sigma notation]] as

:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>

where ''n''! denotes the [[factorial]] of ''n'' and ''ƒ''<sup>&nbsp;(''n'')</sup>(''a'') denotes the ''n''th [[derivative]] of ''ƒ'' evaluated at the point ''a''. The derivative of order zero ''ƒ'' is defined to be ''ƒ'' itself and {{nowrap|(''x'' − ''a'')<sup>0</sup>}} and 0! are both defined to be&nbsp;1. In the case that {{nowrap|''a'' {{=}} 0}}, the series is also called a Maclaurin series.

A Taylor series of ''f'' about point ''a'' may diverge, converge at only the point ''a'', converge for all ''x'' such that <math>|x-a|<R</math> (the largest such ''R'' for which convergence is guaranteed is called the ''radius of convergence''), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at that point.  If the Taylor series at a point has a nonzero [[radius of convergence]], and sums to the function in the [[disc of convergence]], then the function is [[analytic function|analytic]]. The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that the [[exponential function]], the [[logarithm]], the [[trigonometric functions]] and their [[inverse trigonometric functions|inverses]] are extended to functions of a complex variable.

====Fourier series====
{{Main|Fourier series}}
[[Image:Fourier Series.svg|thumb|200px|The first four partial sums of the [[Fourier series]] for a [[Square wave (waveform)|square wave]]. Fourier series are an important tool in real analysis.]]
Fourier series decomposes [[periodic function]]s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely [[sine wave|sines and cosines]]  (or [[complex exponential]]s). The study of Fourier series typically occurs and is handled within the branch [[mathematics]] > [[mathematical analysis]] > [[Fourier analysis]].