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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:

. The first positive integer optional argument is assigned to n.

. The first non-numeric optional argument is assigned to v.

. 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.

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$\mathrm{taylor}\left(\exp \left(x\right)\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5$

$\mathrm{taylor}\left(\exp \left(x\right)\,7\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6$

$\mathrm{taylor}\left(\exp \left(x\right)\,1.0\right)$

$2.718281828x+1.359140914{\left(x-1.0\right)}^2+0.4530469713{\left(x-1.0\right)}^3+0.1132617428{\left(x-1.0\right)}^4+0.02265234857{\left(x-1.0\right)}^5$

$\mathrm{taylor}\left(\frac{1}{v}\exp \left(x\right)\,v\,6\,1\right)$

${\ⅇ}^x-{\ⅇ}^x\left(v-1\right)+{\ⅇ}^x{\left(v-1\right)}^2-{\ⅇ}^x{\left(v-1\right)}^3+{\ⅇ}^x{\left(v-1\right)}^4-{\ⅇ}^x{\left(v-1\right)}^5$

$\mathrm{taylor}\left(\exp \left(x\right)\,7\,6\right)$

${\ⅇ}^6+{\ⅇ}^6\left(x-6\right)+\frac{{\ⅇ}^6{\left(x-6\right)}^2}{2}+\frac{{\ⅇ}^6{\left(x-6\right)}^3}{6}+\frac{{\ⅇ}^6{\left(x-6\right)}^4}{24}+\frac{{\ⅇ}^6{\left(x-6\right)}^5}{120}+\frac{{\ⅇ}^6{\left(x-6\right)}^6}{720}$