basic information - Maple Help

taylor

Taylor series expansion

 Calling Sequence taylor(expression, x=a, n)

Parameters

 expression - expression x - name; independent variable a - real constant; expansion point n - non-negative integer; expansion order

Description

 • The taylor command computes the order n Taylor series expansion of expression, with respect to the variable x, about the point a.

 • The taylor command is thread-safe as of Maple 15.

Examples

 > $\mathrm{taylor}\left(\mathrm{exp}\left(x\right),x=0,4\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}{O}{}\left({{x}}^{{4}}\right)$ (1)
 > $\mathrm{taylor}\left(\frac{1}{x},x=1,3\right)$
 ${1}{-}\left({x}{-}{1}\right){+}{\left({x}{-}{1}\right)}^{{2}}{+}{O}{}\left({\left({x}{-}{1}\right)}^{{3}}\right)$ (2)
 > $\mathrm{taylor}\left(\mathrm{sin}\left(x\right),x=\mathrm{\pi }\right)$
 ${-}\left({x}{-}{\mathrm{\pi }}\right){+}\frac{{1}}{{6}}{}{\left({x}{-}{\mathrm{\pi }}\right)}^{{3}}{-}\frac{{1}}{{120}}{}{\left({x}{-}{\mathrm{\pi }}\right)}^{{5}}{+}{O}{}\left({\left({x}{-}{\mathrm{\pi }}\right)}^{{7}}\right)$ (3)
 > $\mathrm{taylor}\left(\mathrm{sqrt}\left(1+x\right),x,8\right)$
 ${1}{+}\frac{{1}}{{2}}{}{x}{-}\frac{{1}}{{8}}{}{{x}}^{{2}}{+}\frac{{1}}{{16}}{}{{x}}^{{3}}{-}\frac{{5}}{{128}}{}{{x}}^{{4}}{+}\frac{{7}}{{256}}{}{{x}}^{{5}}{-}\frac{{21}}{{1024}}{}{{x}}^{{6}}{+}\frac{{33}}{{2048}}{}{{x}}^{{7}}{+}{O}{}\left({{x}}^{{8}}\right)$ (4)
 > $\mathrm{taylor}\left(\mathrm{sqrt}\left(1+x\right),x=0,8\right)$
 ${1}{+}\frac{{1}}{{2}}{}{x}{-}\frac{{1}}{{8}}{}{{x}}^{{2}}{+}\frac{{1}}{{16}}{}{{x}}^{{3}}{-}\frac{{5}}{{128}}{}{{x}}^{{4}}{+}\frac{{7}}{{256}}{}{{x}}^{{5}}{-}\frac{{21}}{{1024}}{}{{x}}^{{6}}{+}\frac{{33}}{{2048}}{}{{x}}^{{7}}{+}{O}{}\left({{x}}^{{8}}\right)$ (5)

Details

 For detailed information including:
 • Optional parameters
 • Converting a Taylor series to a polynomial
 • The general series command
 see the taylor/details help page.