The commands 1/p and Inverse(p) compute the multiplicative inverse of the power series p. This requires that p is invertible, that is, that p has a non-zero constant term; if that is not the case, an error is signaled.

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.

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

$a≔\mathrm{PowerSeries}\left(1-x-y\right)$

$a≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1-x-y\right]$

$b≔\frac{1}{a}$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}\mathrm{:}1+\mathrm{\dots }\right]$

$c≔\mathrm{Inverse}\left(a\right)$

$c≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b-c\,10\right)$

$0$

$\mathrm{sx}≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d::\mathrm{odd}\,\frac{{\left(\mathrm{-1}\right)}^{\frac{d}{2}-\frac{1}{2}}\cdot x^d}{d!}\,0\right)\,\mathrm{analytic}=\sin \left(x\right)\right)$

$\mathrm{sx}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\sin \left(x\right)\mathrm{:}0+\mathrm{\dots }\right]$

$\frac{1}{\mathrm{sx}}$

Error, not invertible

The MultivariatePowerSeries[Inverse] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.