The series function computes a truncated series 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. If a is infinity, then an asymptotic expansion is computed.

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

Like the series function, when the result is a power series expansion, it is returned using the series type and otherwise it is represented in ordinary sum-of-products form.

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].

The precision n can be infinity. In that case, the computation will terminate either when the input is a polynomial or when it contains a $\mathrm{O}\left(...\right)$ term. This last functionality provides a way to compute an expansion up to a given absolute precision.

The environment variable Order determines the default precision used in intermediate computations when the working precision n is not specified on input.

In rare cases, it might be necessary to increase the value of the global variable Order in order to improve the ability of series 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).

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

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

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

$\mathrm{series}\left(x+\frac{1}{x}\,x=1\,3\right)$

$2+{\left(x-1\right)}^2+O\left({\left(x-1\right)}^3\right)$

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

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

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

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

$\mathrm{series}\left(\mathrm{\Gamma }\left(x\right)\,x=0\,2\right)$

$x^{\mathrm{-1}}-\mathrm{\gamma }+O\left(x\right)$

$\mathrm{series}\left(\frac{x^3}{x^4+4x-5}\,x=\mathrm{\infty }\right)$

$\frac{5}{x^5}-\frac{4}{x^4}+\frac{1}{x}+\mathrm{O}\left(\frac{1}{x^7}\right)$

$\mathrm{series}\left(x^x\,x=0\,3\right)$

$1-\ln \left(\frac{1}{x}\right)x+\frac{1}{2}{\ln \left(\frac{1}{x}\right)}^2{x}^2+O\left(x^3\right)$

$\mathrm{series}\left(\mathrm{sqrt}\left(\sin \left(x\right)\right)\,x=0\,4\right)$

$\sqrt{x}-\frac{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{12}+\mathrm{O}\left(x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{series}\left(\sin \left(x\right)+\mathrm{O}\left(x^{10}\right)\,x\,\mathrm{\infty }\right)$

$x-\frac{1}{6}{x}^3+\frac{1}{120}{x}^5-\frac{1}{5040}{x}^7+\frac{1}{362880}{x}^9+O\left(x^{10}\right)$