Taylor series: Difference between revisions
imported>Charles Blackham No edit summary |
imported>Catherine Woodgold (→General formula: Intuitive explanation of Taylor series) |
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==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 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 | ||
Revision as of 19:34, 30 April 2007
A Taylor series is an infinite sum of polynomial terms to approximate a function in the region about a certain 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} . This is only possible if the function is behaving analytically in this neighbourhood. Such series about the pointFailed 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=0} 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 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 n} .
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 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.
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 }
- 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 (-1 < x \le 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) }