MultivariatePowerSeries/GeometricSeries - Maple Help

MultivariatePowerSeries

 GeometricSeries
 Create the geometric series over a set of variables

 Calling Sequence GeometricSeries(x)

Parameters

 x - variable, or nonempty list of variables

Description

 • GeometricSeries(x) creates the geometric power series in x.
 • If x has more than one element then the result is the geometric series in the sum of the variables in x.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

Examples

 > $\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):$

We create the geometric power series in $x$, that is, the power series for $\frac{1}{1+x}$.

 > $a≔\mathrm{GeometricSeries}\left(x\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}}{:}{1}{+}{x}{+}{\dots }\right]$ (1)

If we truncate $a$ at homogeneous degree 5, this is what we obtain.

 > $\mathrm{Truncate}\left(a,5\right)$
 ${{x}}^{{5}}{+}{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{x}{+}{1}$ (2)

We create the geometric power series in $x+y$, that is, the power series for $\frac{1}{1+x+y}$.

 > $b≔\mathrm{GeometricSeries}\left(\left[x,y\right]\right)$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{\dots }\right]$ (3)

The homogeneous part of $b$ of degree 4 is the following expression.

 > $\mathrm{HomogeneousPart}\left(b,4\right)$
 ${{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}$ (4)

If we truncate $b$ at homogeneous degree 3, this is what we obtain.

 > $\mathrm{Truncate}\left(b,3\right)$
 ${{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{+}{y}{+}{1}$ (5)

Compatibility

 • The MultivariatePowerSeries[GeometricSeries] command was introduced in Maple 2021.