The powseries package contains commands to create and manipulate formal power series represented in general form.

Each command in the powseries package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The following is a list of available commands.

Other trigonometric functions, such as tan(x), sec(x), and the hyperbolic functions, such as sinh(x), can be generated through the powseries:-evalpow command.

To display the help page for a particular powseries command, see Getting Help with a Command in a Package.

For more information on the general representation of formal power series, see Representation of a Formal Power Series.

$a≔\mathrm{powseries}:-\mathrm{powexp}\left(x\right)\:$

$b≔\mathrm{powseries}:-\mathrm{tpsform}\left(a\,x\,5\right)$

$b≔1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+O\left(x^5\right)$

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

$c≔\mathrm{powadd}\left(\mathrm{powpoly}\left(1+x^2+x\,x\right)\,\mathrm{powlog}\left(1+x\right)\right)\:$

$d≔\mathrm{tpsform}\left(c\,x\,6\right)$

$d≔1+2x+\frac{1}{2}{x}^2+\frac{1}{3}{x}^3-\frac{1}{4}{x}^4+\frac{1}{5}{x}^5+O\left(x^6\right)$

$e≔\mathrm{evalpow}\left(\sinh \left(x\right)\right)\:$

$\mathrm{tpsform}\left(e\,x\,5\right)$

$x+\frac{1}{6}{x}^3+O\left(x^5\right)$

$g≔\mathrm{tpsform}\left(\mathrm{powdiff}\left(\mathrm{powsin}\left(x\right)\right)\,x\,6\right)$

$g≔1-\frac{1}{2}{x}^2+\frac{1}{24}{x}^4+O\left(x^6\right)$

$h≔\mathrm{evalpow}\left(\mathrm{Tan}\left(x\right)\right)\:$

$k≔\mathrm{tpsform}\left(\mathrm{negative}\left(h\right)\,x\,5\right)$

$k≔-x-\frac{1}{3}{x}^3+O\left(x^5\right)$