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'''Taylor series''' are an infinite sum of polynomial terms to approximate a function in the region about a certain point <math>a</math>. This is only possible is the function is behaving analytically in this neighbourhood. Such series about the point<math>a=0</math> are known as '''Maclaurin series''', after Scottish mathematician Colin Maclaurin. They work by ensuring that the approximate series matches the n^th derivative of the function being approximated when it is approximated by a polynomial of degree n.
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A '''Taylor series''' is an infinite sum of polynomial terms to approximate a function in the region about a certain point <math>a</math>. This is only possible if the function is behaving analytically in this neighbourhood. Such series about the point <math>a=0</math> are known as '''Maclaurin series''', after Scottish mathematician Colin Maclaurin. They work by ensuring that the approximate series matches up to the n<sup>th</sup> derivative of the function being approximated when it is approximated by a polynomial of degree <math>n</math>.


==Proof==
==Proof==
Line 6: Line 8:
==Series==
==Series==
===General formula===
===General formula===
An intuitive explanation of the Taylor series is that, in order to approximate the value of <math>f(x)</math>, as a first approximation we use the value at another point <math>a</math>, i.e. <math>f(a)</math>.  If <math>x</math> and <math>a</math> are close together and <math>f</math> varies only slowly, this can be a good approximation.  Then we refine the approximation step by step.  The derivative of <math>f</math> is used to calculate approximately how much <math>f</math> would be expected to change between <math>a</math> and <math>x</math>, and this amount is added as a correction.  But we assume we only know the derivative of <math>f</math> at <math>a</math>, and the derivative may change between the two numbers, so another correction is needed, involving the second derivative which is a measure of how much the first derivative changes.  So it continues, adding corrections to corrections, and in the limit, if it converges then it converges to the actual value of <math>f(x)</math> even if <math>x</math> and <math>a</math> are far apart.
:<math>
:<math>
f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots+\frac{f^{(r)}(a)}{r!}(x-a)^r+\cdots  
f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots+\frac{f^{(r)}(a)}{r!}(x-a)^r+\cdots  
</math>
</math>
::<math>=\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}</math>
::<math>=\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}</math>
where <math>\scriptstyle f'</math> is the first derivative of the function <math>\scriptstyle f</math>, and <math>\scriptstyle f''</math> is the second derivative, and so on. Most practically used smooth functions satisfy this formula but not all of them. The smooth function <math>f(x)=\exp(-1/x^2)</math> have all derivatives equal to zero but the function itself is not equal to zero. In order to prove Taylor formula for particular functions, mathematicians reduce the problem to proving Taylor formula for basic functions using a variety of theorems. Taylor formula for basic functions can be proved directly by estimating their derivatives.
==Convergence of the Taylor series and examples==
Region of convergence of the Taylor series is always circle. This circle is centered at the point of expansion.
Radius of this circle is equal to the distance from this point  to the closest singulatiry. Within the circle of convergence,
the Taylor series can be used to make the [[polynomial approximation]] of [[holomorphic function]]s, just truncating the series.
=== f(z)=1/z ===
{{Image|TaylorExampleZ.jpg|right|600px|Function <math>f=1/z</math> and its approximation made of the Taylor series at point <math>z=1</math> in the complex <math>z</math> plane.}}
The simple function, which is not polynomoal, is <math>f=1/z</math> This function is shown in the left hand side of the figure in the complex <math>z</math> plane with lines of constant <math>\Re(f)</math> and constant <math>\Im(f)</math>. The polynomial approximation of this function can be obtained truncating the Taylor series; for example, taking 21 terms of the Taylor series developed at <math>z=1</math> gives the approximation
:<math> t_{20}(z)=\sum_{n=0}^{20} (1-z)^n</math>
This function is shown in the right hand side of the figure. In the central part of the circle of convergence, the
truncated Taylor series approximate the function.
Below, more example of Taylor series of various functions are suggested.
===Exponential & Logarithmic functions===
===Exponential & Logarithmic functions===
:<math>
:<math>
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^r}{r!}+\cdots \qquad \forall x  
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^r}{r!}+\cdots \qquad \forall x \in \mathbb{C}
</math><br/><br/>
</math><br/><br/>
:<math>
:<math>
ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{r+1}\frac{x^r}{r}+\cdots \qquad (-1 < x \le 1)
\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{r+1}\frac{x^r}{r}+\cdots \qquad  
\forall x \in \mathbb{C}: |x|\le 1 , x\ne -1
</math>
</math>
===Trigonometric functions===
===Trigonometric functions===
:<math>
:<math>
\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+(-1)^{r}\frac{x^{2r+1}}{(2r+1)!}+\cdots \qquad \forall x  
\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+(-1)^{r}\frac{x^{2r+1}}{(2r+1)!}+\cdots \qquad \forall x  
</math><br/><br/>
</math><br/><br/>
:<math>
:<math>
\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+(-1)^{r}\frac{x^{2r}}{(2r)!}+\cdots \qquad \forall x  
\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+(-1)^{r}\frac{x^{2r}}{(2r)!}+\cdots \qquad \forall x  
</math><br/><br/>
</math><br/><br/>
:<math>
\tan x=x+\frac{x^3}{3}+\frac{2 x^5}{15}+\cdots+\frac{B_{2r} (-4)^r (1-4^r)}{(2r)!} x^{2r-1}+\cdots \qquad |x|<\frac{\pi}{2}
</math><br/>where ''B<sub>k</sub>''=k<sup>th</sup> [[Bernoulli number]].
<br/><br/>
===Inverse trigonometric functions===
===Inverse trigonometric functions===
:<math>
\operatorname{tan^{-1}} x=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots+(-1)^{r}\frac{x^{2r+1}}{(2r+1)}+\cdots \qquad (-1 < x \le 1)
</math><br/><br/>
===Hyperbolic functions===
===Hyperbolic functions===
:<math>
\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+\frac{x^{2r+1}}{(2r+1)!}+\cdots \qquad \forall x
</math><br/><br/>
:<math>
\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+\frac{x^{2r}}{(2r)!}+\cdots \qquad \forall x
</math><br/><br/>
===Inverse hyperbolic functions===
===Inverse hyperbolic functions===
:<math>
\operatorname{tanh^{-1}} x=x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots+\frac{x^{2r+1}}{(2r+1)}+\cdots \qquad (-1 < x \le 1)
</math><br/><br/>
==Calculation of Taylor series for more complicated functions==


 
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A Taylor series is an infinite sum of polynomial terms to approximate a function in the region about a certain point . This is only possible if the function is behaving analytically in this neighbourhood. Such series about the point are known as Maclaurin series, after Scottish mathematician Colin Maclaurin. They work by ensuring that the approximate series matches up to the nth derivative of the function being approximated when it is approximated by a polynomial of degree .

Proof

See Taylor's theorem

Series

General formula

An intuitive explanation of the Taylor series is that, in order to approximate the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , as a first approximation we use the value at another point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} are close together and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} varies only slowly, this can be a good approximation. Then we refine the approximation step by step. The derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is used to calculate approximately how much Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} would be expected to change between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , and this amount is added as a correction. But we assume we only know the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , and the derivative may change between the two numbers, so another correction is needed, involving the second derivative which is a measure of how much the first derivative changes. So it continues, adding corrections to corrections, and in the limit, if it converges then it converges to the actual value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} even if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} are far apart.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots+\frac{f^{(r)}(a)}{r!}(x-a)^r+\cdots }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f'} is the first derivative of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f''} is the second derivative, and so on. Most practically used smooth functions satisfy this formula but not all of them. The smooth function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\exp(-1/x^2)} have all derivatives equal to zero but the function itself is not equal to zero. In order to prove Taylor formula for particular functions, mathematicians reduce the problem to proving Taylor formula for basic functions using a variety of theorems. Taylor formula for basic functions can be proved directly by estimating their derivatives.

Convergence of the Taylor series and examples

Region of convergence of the Taylor series is always circle. This circle is centered at the point of expansion. Radius of this circle is equal to the distance from this point to the closest singulatiry. Within the circle of convergence, the Taylor series can be used to make the polynomial approximation of holomorphic functions, just truncating the series.

f(z)=1/z

Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=1/z} and its approximation made of the Taylor series at point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=1} in the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} plane.

The simple function, which is not polynomoal, is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=1/z} This function is shown in the left hand side of the figure in the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} plane with lines of constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(f)} and constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(f)} . The polynomial approximation of this function can be obtained truncating the Taylor series; for example, taking 21 terms of the Taylor series developed at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=1} gives the approximation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{20}(z)=\sum_{n=0}^{20} (1-z)^n}

This function is shown in the right hand side of the figure. In the central part of the circle of convergence, the truncated Taylor series approximate the function.

Below, more example of Taylor series of various functions are suggested.

Exponential & Logarithmic functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^r}{r!}+\cdots \qquad \forall x \in \mathbb{C} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{r+1}\frac{x^r}{r}+\cdots \qquad \forall x \in \mathbb{C}: |x|\le 1 , x\ne -1 }

Trigonometric functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+(-1)^{r}\frac{x^{2r+1}}{(2r+1)!}+\cdots \qquad \forall x }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+(-1)^{r}\frac{x^{2r}}{(2r)!}+\cdots \qquad \forall x }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan x=x+\frac{x^3}{3}+\frac{2 x^5}{15}+\cdots+\frac{B_{2r} (-4)^r (1-4^r)}{(2r)!} x^{2r-1}+\cdots \qquad |x|<\frac{\pi}{2} }
where Bk=kth Bernoulli number.



Inverse trigonometric functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{tan^{-1}} x=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots+(-1)^{r}\frac{x^{2r+1}}{(2r+1)}+\cdots \qquad (-1 < x \le 1) }

Hyperbolic functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+\frac{x^{2r+1}}{(2r+1)!}+\cdots \qquad \forall x }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+\frac{x^{2r}}{(2r)!}+\cdots \qquad \forall x }

Inverse hyperbolic functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{tanh^{-1}} x=x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots+\frac{x^{2r+1}}{(2r+1)}+\cdots \qquad (-1 < x \le 1) }

Calculation of Taylor series for more complicated functions

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