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[[Image:Exp series.gif|right|thumb|An [[animation]] that shows how a Taylor series can be used to [[approximate]] a function. The blue line shows the [[exponential function]] <math>f(x)=e^{x}</math>. The red lines show the [[sum]] of ''n'' [[derivatives]] -- that is, ''n''+1 [[term]]s in the Taylor series. As ''n'' gets bigger, the red line gets closer to the blue line.]]

[[File:Exp series.gif|right|thumb|An [[animation]] that shows how a Taylor series can be used to [[approximate]] a function. The blue line shows the [[exponential function]] <math>f(x)=e^{x}</math>. The red lines show the [[sum]] of ''n'' [[derivatives]] -- that is, ''n''+1 [[term]]s in the Taylor series. As ''n'' gets bigger, the red line gets closer to the blue line.]]

A '''Taylor series''' is an  idea used in [[computer science]], [[calculus]], and other kinds of higher-level [[mathematics]]. It is a [[Series (mathematics)|series]] that is used to create an [[estimate]] (guess) of what a [[Function (mathematics)|function]] looks like. There is also a special kind of Taylor series called a '''Maclaurin series'''.

A '''Taylor series''' is an  idea used in [[computer science]], [[calculus]], chemistry, physics and other kinds of higher-level [[mathematics]]. It is a [[Series (mathematics)|series]] that is used to create an [[estimate]] (guess) of what a [[Function (mathematics)|function]] looks like. There is also a special kind of Taylor series called a '''Maclaurin series'''.

The theory behind the Taylor series is that if a point is chosen on the [[coordinate plane]] ([[x-axis|x-]] and [[y-axis|y-axes]]), then it is possible to guess what a function will look like in the area around that point. This is done by taking the [[derivative]]s of the function and adding them all together. The idea is that it is possible to add the [[infinite]] number of derivatives and come up with a single [[finite]] sum.

The theory behind the Taylor series is that if a point is chosen on the [[coordinate plane]] ([[x-axis|x-]] and [[y-axis|y-axes]]), then it is possible to guess what a function will look like in the area around that point. This is done by taking the [[Derivative (mathematics)|derivative]]s of the function and adding them all together. The idea is that it is possible to add the [[infinite]] number of derivatives and come up with a single [[finite]] sum.

In [[mathematics]], a Taylor series shows a [[Function (mathematics)|function]] as an [[infinite sum]]. The sum's terms are taken from the function's [[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 "[[Zeno's paradox]]" was 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 parodox'] 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.

Another Greek philosopher, [[Aristotle]], came up with an answer to the philosophical question. It was [[Archimedes]], however, who came up with a mathematical solution using his [[method of exhaustion]]. He was able to prove that when something is split up into an infinite number of tiny pieces, they will still add up to a single whole when all of them are added back together.<ref>{{cite book|last=Kline|first=M|year=1990|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|pages=35-37}}</ref> The ancient [[China|Chinese]] [[mathematician]] [[Liu Hui]] proved the same thing several hundred years later.<ref>{{cite book|last1=Boyer|first1=C|last2=Merzbach|first2=U|year=1991|title=A History of Mathematics|publisher=John Wiley and Sons|pages=202-203}}</ref>

Another Greek philosopher, [[Aristotle]], came up with an answer to the philosophical question. It was [[Archimedes]], however, who came up with a mathematical solution using his [[method of exhaustion]]. He was able to prove that when something is split up into an infinite number of tiny pieces, they will still add up to a single whole when all of them are added back together.<ref>{{cite book|last=Kline|first=M|year=1990|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|pages=35-37}}</ref> The ancient [[China|Chinese]] [[mathematician]] [[Liu Hui]] proved the same thing several hundred years later.<ref>{{cite book|last1=Boyer|first1=C|last2=Merzbach|first2=U|year=1991|title=A History of Mathematics|publisher=John Wiley and Sons|pages=202-203}}</ref>

The earliest known examples of the Taylor series are the work of [[Mādhava of Sañgamāgrama]] in [[India]] in the 1300s.<ref name="MAT 314">{{cite web| publisher=Canisius College| work=MAT 314|url=http://www.canisius.edu/topos/rajeev.asp| title=Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala| accessdate=2006-07-09}}</ref> Later [[Indian mathematics|Indian mathematicians]] wrote about his work with the [[trigonometric functions]] of [[sine]], [[cosine]], [[tangent]], and [[arctangent]]. None of Mādhava's writings or records still exist today. Other mathematicians based their work on Mādhava's discoveries and worked more with these series until the 1500s.

The earliest known examples of the Taylor series are the work of [[Mādhava of Sañgamāgrama]] in [[India]] in the 1300s.<ref name="MAT 314">{{cite web| publisher=Canisius College| work=MAT 314|url=http://www.canisius.edu/topos/rajeev.asp| title=Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala| accessdate=2006-07-09}}</ref> Later [[Indian mathematics|Indian mathematicians]] wrote about his work with the [[trigonometric functions]] of [[sine]], [[cosine]], [[tangent]], and [[arctangent]]. None of Mādhava's writings or records still exist today. Other mathematicians based their work on Mādhava's discoveries and worked more with these series until the 1500s.

[[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|publisher=Harvard University Press|location=Cambridge, Massachusetts|year=1969|pages=329-332}}</ref> [[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|publisher=Harvard University Press|location=Cambridge, Massachusetts|year=1969|pages=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 ==

Some important Taylor series and Maclaurin series are the following.

Some important Taylor series and Maclaurin series are the following.

:<math>\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x\!</math>

:<math>\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x\!</math>

:<math>\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\text{ for all } x\!</math>

:<math>\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\text{ for all } x\!</math>

:<math>\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\text{ for }|x| < \frac{\pi}{2}\!</math>

:<math>\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\text{ for }|x| < \frac{\pi}{2}\!</math>

== References ==

 

== References cite 3 is missing.  ==

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[[Category:Mathematics]]

[[Category:Calculus]]

 

[[pl:Wzór Taylora#Szereg Taylora]]

[[pl:Wzór Taylora#Szereg Taylora]]