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 ApproximatelyZero(p) returns true if the power series is equal to zero up to its currently computed precision, otherwise false.

The command ApproximatelyZero(p,deg) returns true,  if the power series is equal to zero up to precision deg, otherwise false.

The command ApproximatelyZero(u) returns true if each coefficient of the univariate polynomial over power series is equal to zero up to its currently computed precision, otherwise false.

The command ApproximatelyZero(u,deg) returns true if each coefficient of the univariate polynomial over power series is equal to zero 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.

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

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

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

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

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

$c≔a-b$

$c≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}0\right]$

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

$\mathrm{false}$

$\mathrm{ApproximatelyZero}\left(c\&comma;10\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(a\&comma;b\&comma;10\right)$

$\mathrm{true}$

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

$f≔\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

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

$g≔\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}\left(0\right)\mathrm{+}\left(1-x-y\right)z\right]$

$h≔f-g$

$h≔\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}\left(0\right)\mathrm{+}\left(0\right)z\right]$

$\mathrm{ApproximatelyZero}\left(h\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyZero}\left(h\&comma;10\right)$

$\mathrm{true}$

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

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