Two power series p and q are said to be equal up to a degree deg, called the precision, if for each degree d <= deg, the homogeneous components of p and q of degree d are equal.

The command ApproximatelyEqual(p,q) returns true if the two power series are equal up to the minimum of their currently computed precisions, otherwise false.

The command  ApproximatelyEqual(p,q,deg) returns true if the two power series are equal up to precision deg, otherwise false. This calling sequence will compute any coefficients needed that haven't been computed so far.

The command  ApproximatelyEqual(u,v) returns true if the two univariate polynomials over power series are equal up to the currently computed precision of each coefficient power series, otherwise false.

The command  ApproximatelyEqual(u,v,deg) returns true, if the two univariate polynomials over power series are equal by comparing each power series coefficient up to precision deg, otherwise false.

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.

$\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right)\&colon;$

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

$a≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{1}{1+x+y}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(a\&comma;1\right)$

$1-x-y$

$b≔\mathrm{GeometricSeries}\left(\left[x\&comma;y\right]\right)$

$b≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{1}{1-x-y}\mathrm{:}1+x+y+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b\&comma;1\right)$

$1+x+y$

$c≔\mathrm{Inverse}\left(\mathrm{SumOfAllMonomials}\left(\left[x\&comma;y\right]\right)\right)$

$c≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(c\&comma;1\right)$

$1-x-y$

$\mathrm{ApproximatelyEqual}\left(a\&comma;b\right)$

$\mathrm{false}$

$\mathrm{ApproximatelyEqual}\left(a\&comma;c\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(a\&comma;c\&comma;2\right)$

$\mathrm{false}$

$\mathrm{Truncate}\left(a\right)$

$x^2+2xy+y^2-x-y+1$

$\mathrm{Truncate}\left(c\right)$

$xy-x-y+1$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right)\&comma;\mathrm{GeometricSeries}\left(\left[x\&comma;y\right]\right)\right]\&comma;z\right)\&colon;$

$g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right)\&comma;\mathrm{Inverse}\left(\mathrm{PowerSeries}\left(1-x-y\right)\right)\right]\&comma;z\right)\&colon;$

$\mathrm{ApproximatelyEqual}\left(f\&comma;g\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(f\&comma;g\&comma;10\right)$

$\mathrm{true}$

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

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