992,540
edits
Changes
m
Fixing links to disambiguation pages using AWB
[[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.]]
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'''.
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.
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]].
== 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.
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.
[[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''.
== 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''.
This Taylor series, written as a [[power series]], looks like:
:<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>
This formula can also be written in [[sigma notation]] as:
:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
Here ''n''! is the [[factorial]] of ''n''. ''ƒ''<sup> (''n'')</sup>(''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 ===
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>
When written in [[sigma notation]], the Maclaurin series is:
:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(0)}{n!} \, x^{n}</math>
== Common Taylor series ==
Some important Taylor series and Maclaurin series are the following.
[[Trigonometric function]]s:
:<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>\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 ==
{{reflist}}
[[Category:Mathematics]]
{{Link GA|en}}
[[pl:Wzór Taylora#Szereg Taylora]]
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'''.
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.
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]].
== 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.
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.
[[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''.
== 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''.
This Taylor series, written as a [[power series]], looks like:
:<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>
This formula can also be written in [[sigma notation]] as:
:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
Here ''n''! is the [[factorial]] of ''n''. ''ƒ''<sup> (''n'')</sup>(''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 ===
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>
When written in [[sigma notation]], the Maclaurin series is:
:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(0)}{n!} \, x^{n}</math>
== Common Taylor series ==
Some important Taylor series and Maclaurin series are the following.
[[Trigonometric function]]s:
:<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>\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 ==
{{reflist}}
[[Category:Mathematics]]
{{Link GA|en}}
[[pl:Wzór Taylora#Szereg Taylora]]