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.

By default, if p knows its analytic expression and this analytic expression is zero, the ApproximatelyZero command will immediately return true. If you pass the force or force = true option, then the ApproximatelyZero command will ignore the analytic expressions. The default behavior can be explicitly selected by passing the option force = false.

Any form of the force option passed to the ApproximatelyZero command when comparing a univariate polynomial over power series are used as described above when comparing the power series coefficients.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

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

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

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

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

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

$c≔a-b$

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

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

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

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

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

$h≔f-g$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

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

The MultivariatePowerSeries[ApproximatelyZero] command was updated in Maple 2022.

The force option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.