The command Truncate(p, d) returns the sum of all homogeneous parts of p of degree at most d. In other words, Truncate(p, d) returns  the natural image of the power series p modulo  the ${\left(d\+1\right)}^{\mathrm{th}}$ power of the maximal ideal of the ring of power series.

If d is greater than the current precision of p, then the necessary extra terms are computed.

If d is not specified, all currently computed terms are used and no extra ones are computed. In other words, the default value of d is the precision of p.

The command  Truncate(u, d) returns the polynomial obtained from u by replacing each coefficient p with the sum of the homogeneous parts of p of degree at most d. In other words, each coefficient is replaced by its image modulo the ${\left(d\+1\right)}^{\mathrm{th}}$ power of the maximal ideal of the ring of power series.

If d is not specified, the precision of each coefficient is used. If the coefficients are currently known to different precisions, they will consequently be truncated at different degrees.

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.

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

$a≔\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)$

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

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

$1+x+y$

$\mathrm{UpdatePrecision}\left(a\,3\right)$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}\mathrm{:}1+x+y+x^2+2xy+y^2+x^3+3x^{2}y+3x{y}^2+y^3+\mathrm{\dots }\right]$

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

$x^3+3x^{2}y+3x{y}^2+y^3+x^2+2xy+y^2+x+y+1$

$\mathrm{Truncate}\left(a\,2\right)$

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

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(x\right)\,\mathrm{GeometricSeries}\left(y\right)\,a\right]\,z\right)$

$f≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1+x+\mathrm{\dots }\right)\mathrm{+}\left(1+y+\mathrm{\dots }\right)z\mathrm{+}\left(1+x+y+x^2+2xy+y^2+x^3+3x^{2}y+3x{y}^2+y^3+\mathrm{\dots }\right)z^2\right]$

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

$\left(x^3+3x^{2}y+3x{y}^2+y^3+x^2+2xy+y^2+x+y+1\right)z^2+\left(1+y\right)z+1+x$

$\mathrm{Truncate}\left(f\,2\right)$

$\left(x^2+2xy+y^2+x+y+1\right)z^2+\left(y^2+y+1\right)z+x^2+x+1$

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

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