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In [[mathematics]], a Taylor series shows a [[Function (mathematics)|function]] as the sum of an [[wikipedia:Series_(mathematics)|infinite series]]. The sum's terms are taken from the function's [[Derivative (mathematics)|derivative]]s. Taylor series come from [[Taylor's theorem]].

In [[mathematics]], a Taylor series shows a [[Function (mathematics)|function]] as the sum of an [[wikipedia:Series_(mathematics)|infinite series]]. The sum's terms are taken from the function's [[Derivative (mathematics)|derivative]]s. Taylor series come from [[Taylor's theorem]].

== History ==

== History ==

The [[Ancient Greece|Ancient Greek]] [[philosopher]] [[Zeno of Elea]] first came up with the idea of this series. The [[paradox]] called "[https://en.m.wikipedia.org/wiki/Zeno's_paradoxes Zeno's paradox]" is the result. He believed that it would be impossible to add an [[infinity|infinite]] number of values and get a single [[finite]] value as a result.

The [[Ancient Greece|Ancient Greek]] [[philosopher]] [[Zeno of Elea]] first came up with the idea of this series. The [[paradox]] called "[https://en.m.wikipedia.org/wiki/Zeno's_paradoxes Zeno's paradox]" is the result. He believed that it would be impossible to add an [[infinity|infinite]] number of values and get a single [[finite]] value as a result.

[[James Gregory (mathematician)|James Gregory]], a [[Scotland|Scottish]] mathematician, worked in this area in the 1600s. Gregory studied the Taylor series and published several Maclaurin series. In 1715, [[Brook Taylor]] discovered a general method for applying the series to all [[function (mathematics)|functions]]. (All of the previous research showed how to apply the method to only specific functions.)<ref>{{cite book|last=Taylor|first=Brook|title=Methodus Incrementorum Directa et Inversa|language=Latin|location=London|year=1715|chapter=Proposition VII, Theorem 3, Corollary 2|pages=21–23}} ''cited in'' {{cite book|last=Struik|first=D.J.|title=A Source Book in Mathematics 1200-1800|url=https://archive.org/details/sourcebookinmath0000stru|publisher=Harvard University Press|location=Cambridge, Massachusetts|year=1969|pages=[https://archive.org/details/sourcebookinmath0000stru/page/329 329]–332}}</ref> [[:en:Colin_Maclaurin|Colin Maclaurin]] published a special case of the Taylor series in the 1700s. This series, which is based around [[zero]], is called the ''Maclaurin series''.

[[James Gregory (mathematician)|James Gregory]], a [[Scotland|Scottish]] mathematician, worked in this area in the 1600s. Gregory studied the Taylor series and published several Maclaurin series. In 1715, [[Brook Taylor]] discovered a general method for applying the series to all [[function (mathematics)|functions]]. (All of the previous research showed how to apply the method to only specific functions.)<ref>{{cite book|last=Taylor|first=Brook|title=Methodus Incrementorum Directa et Inversa|language=Latin|location=London|year=1715|chapter=Proposition VII, Theorem 3, Corollary 2|pages=21–23}} ''cited in'' {{cite book|last=Struik|first=D.J.|title=A Source Book in Mathematics 1200-1800|url=https://archive.org/details/sourcebookinmath0000stru|publisher=Harvard University Press|location=Cambridge, Massachusetts|year=1969|pages=[https://archive.org/details/sourcebookinmath0000stru/page/329 329]–332}}</ref> [[:en:Colin_Maclaurin|Colin Maclaurin]] published a special case of the Taylor series in the 1700s. This series, which is based around [[zero]], is called the ''Maclaurin series''.

== Definition ==

== Definition ==

A Taylor series can be used to describe any function ''ƒ''(''x'') that is a [[smooth function]] (or, in mathematical terms, "infinitely differentiable.") The function ''ƒ'' can be either [[real number|real]] or [[complex number|complex]]. The Taylor series is then used to describe what the function looks like in the [[neighborhood (mathematics)|neighborhood]] of some number ''a''.

A Taylor series can be used to describe any function ''ƒ''(''x'') that is a [[smooth function]] (or, in mathematical terms, "infinitely differentiable.") The function ''ƒ'' can be either [[real number|real]] or [[complex number|complex]]. The Taylor series is then used to describe what the function looks like in the [[neighborhood (mathematics)|neighborhood]] of some number ''a''.

Here ''n''! is the [[factorial]] of ''n''. ''ƒ''^{&nbsp;(''n'')}(''a'') is the ''n''th [[derivative]] of ''ƒ'' at the point ''a''. <math>a</math> is a number in the function's [[domain]]. If the Taylor Series of a function is equal to that function, the function is called an "[[analytic]] function."

Here ''n''! is the [[factorial]] of ''n''. ''ƒ''^{&nbsp;(''n'')}(''a'') is the ''n''th [[derivative]] of ''ƒ'' at the point ''a''. <math>a</math> is a number in the function's [[domain]]. If the Taylor Series of a function is equal to that function, the function is called an "[[analytic]] function."

 

== Maclaurin series ==

=== Maclaurin series ===

When <math>a=0</math>, the function is called a '''Maclaurin series'''. The Maclaurin series written as a [[power series]] looks like:

When <math>a=0</math>, the function is called a '''Maclaurin series'''. The Maclaurin series written as a [[power series]] looks like:

:<math>f(0)+\frac {f'(0)}{1!} x+ \frac{f''(0)}{2!} x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots. </math>

:<math>f(0)+\frac {f'(0)}{1!} x+ \frac{f''(0)}{2!} x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots. </math>

When written in [[sigma notation]], the Maclaurin series is:

When written in [[sigma notation]], the Maclaurin series is:

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

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

== Common Taylor series ==

== Common Taylor series ==

Some important Taylor series and Maclaurin series are the following.

Some important Taylor series and Maclaurin series are the following.

Where <math>B_{n}</math> is the nth [[Bernoulli number]], and <math>\ln</math> is the [[natural logarithm]].

Where <math>B_{n}</math> is the nth [[Bernoulli number]], and <math>\ln</math> is the [[natural logarithm]].

== References ==

== References ==

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