Editing Real analysis (section)

====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]]
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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.