The taylor function computes a truncated Taylor expansion of expr, with respect to the variable x, about the point a, up to order n. If a is not given, it defaults to 0.

The taylor function of the MultiSeries package is intended to be used in the same manner as the top-level taylor function.

If the given expression does not have a Taylor expansion around a, then taylor issues an error. In that case, the MultiSeries[series] or MultiSeries[multiseries] functions can be used to obtain a more general series expansion.

The underlying engine for computing expansions is the MultiSeries[multiseries] function. In particular, the variable x is assumed to tend to its limit point a in the manner described in MultiSeries[multiseries].

In rare cases, it might be necessary to increase the value of the global variable Order in order to improve the ability of taylor to solve problems with significant cancellation. This is made explicit by an error message coming from multiseries.

It can also happen that the result is wrong because Testzero failed to recognize that the leading coefficient of a multiseries expansion happens to be 0. In those cases, it is necessary to modify this environment variable (see Testzero).

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

$\mathrm{taylor}\left(\frac{x}{1-x-x^2}\,x=0\right)$

$x+x^2+2x^3+3x^4+5x^5+O\left(x^6\right)$

$\mathrm{taylor}\left(\exp \left(x\right)\,x=0\,8\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6+\frac{1}{5040}{x}^7+O\left(x^8\right)$

$\mathrm{taylor}\left(\frac{\cos \left(x\right)}{x}\,x\right)$

Error, (in MultiSeries:-taylor) does not have a taylor expansion, try series()

$\mathrm{series}\left(\frac{\cos \left(x\right)}{x}\,x\right)$

$x^{\mathrm{-1}}-\frac{1}{2}x+\frac{1}{24}{x}^3-\frac{1}{720}{x}^5+O\left(x^6\right)$