 taylor - Maple Help

MTM

 taylor
 Taylor series expansion Calling Sequence taylor(f,n,v,a) Parameters

 f - expression n - (optional) positive integer v - (optional) symbol a - (optional) real constant Description

 • The function taylor(f,n,v,a) interprets the expression f as a function of v, and returns the (n-1)-degree Taylor series expansion of f(v) about the point a.
 • If n is not specified, then n = 6.
 • If v is not specified, then v is the default symbol of f, given by findsym(f,1).
 • If a is not specified, then a = 0.
 • The optional parameters need not be passed in the order, n,v,a. More precisely, the arguments are assigned to n, v, and a in the following way:
 I. . The first positive integer optional argument is assigned to n.
 II. . The first non-numeric optional argument is assigned to v.
 III. . The first real numeric optional argument is assigned to a, unless the argument is also a positive integer and n has not yet been assigned a value. Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $\mathrm{taylor}\left(\mathrm{exp}\left(x\right)\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}$ (1)
 > $\mathrm{taylor}\left(\mathrm{exp}\left(x\right),7\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{+}\frac{{1}}{{720}}{}{{x}}^{{6}}$ (2)
 > $\mathrm{taylor}\left(\mathrm{exp}\left(x\right),1.0\right)$
 ${2.718281828}{}{x}{+}{1.359140914}{}{\left({x}{-}{1.0}\right)}^{{2}}{+}{0.4530469713}{}{\left({x}{-}{1.0}\right)}^{{3}}{+}{0.1132617428}{}{\left({x}{-}{1.0}\right)}^{{4}}{+}{0.02265234857}{}{\left({x}{-}{1.0}\right)}^{{5}}$ (3)
 > $\mathrm{taylor}\left(\frac{1}{v}\mathrm{exp}\left(x\right),v,6,1\right)$
 ${{ⅇ}}^{{x}}{-}{{ⅇ}}^{{x}}{}\left({v}{-}{1}\right){+}{{ⅇ}}^{{x}}{}{\left({v}{-}{1}\right)}^{{2}}{-}{{ⅇ}}^{{x}}{}{\left({v}{-}{1}\right)}^{{3}}{+}{{ⅇ}}^{{x}}{}{\left({v}{-}{1}\right)}^{{4}}{-}{{ⅇ}}^{{x}}{}{\left({v}{-}{1}\right)}^{{5}}$ (4)
 > $\mathrm{taylor}\left(\mathrm{exp}\left(x\right),7,6\right)$
 ${{ⅇ}}^{{6}}{+}{{ⅇ}}^{{6}}{}\left({x}{-}{6}\right){+}\frac{{{ⅇ}}^{{6}}{}{\left({x}{-}{6}\right)}^{{2}}}{{2}}{+}\frac{{{ⅇ}}^{{6}}{}{\left({x}{-}{6}\right)}^{{3}}}{{6}}{+}\frac{{{ⅇ}}^{{6}}{}{\left({x}{-}{6}\right)}^{{4}}}{{24}}{+}\frac{{{ⅇ}}^{{6}}{}{\left({x}{-}{6}\right)}^{{5}}}{{120}}{+}\frac{{{ⅇ}}^{{6}}{}{\left({x}{-}{6}\right)}^{{6}}}{{720}}$ (5)