Taylor series: Difference between revisions

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imported>Charles Blackham
(structure, hyp f'ns)
imported>Charles Blackham
(inv tan (circ & hyp))
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</math><br/><br/>
</math><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>
:<math>
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</math><br/><br/>
</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==
==Calculation of Taylor series for more complicated functions==


[[Category:CZ Live]]
[[Category:CZ Live]]
[[Category:Mathematics Workgroup]]
[[Category:Mathematics Workgroup]]

Revision as of 03:49, 27 April 2007

Taylor series are an infinite sum of polynomial terms to approximate a function in the region about a certain point . This is only possible is 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 the n^th derivative of the function being approximated when it is approximated by a polynomial of degree n.

Proof

See Taylor's theorem

Series

General formula

Exponential & Logarithmic functions



Trigonometric functions





Inverse trigonometric functions



Hyperbolic functions





Inverse hyperbolic functions



Calculation of Taylor series for more complicated functions