The command p1 + p2 returns the sum of the terms p1 and p2. The result is a power series.

The command Add(P) returns the sum of the terms in P.

The command Add(P, coefficients = C) returns the sum of the products C[i] * P[i]. The length of the list C must be the same as the number of elements of P.

The command u1 + u2 returns the sum of the terms u1 and u2. The result is a univariate polynomial over power series.

The command Add(U) returns the sum of the entries of U. They are converted to univariate polynomials over power series in the same variable. If this is not possible, an error is raised. This may happen if there are univariate polynomials over power series in different variables. It can also happen if the univariate polynomials over power series all have the same main variable, say x, but one of the other arguments is a power series that is not known to be expressible as a polynomial in x. The same restrictions apply to the calling sequence u1 + u2.

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.

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

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

$b≔\mathrm{PowerSeries}\left(1+x+y+z\right)\:$

$c≔\mathrm{PowerSeries}\left(2xy+3z^3\right)\:$

$a+b$

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

$a+1$

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

$\mathrm{Add}\left(a\,b\,c\,1+xyz\right)$

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

$\mathrm{Add}\left(a\,b\,c\,\mathrm{coefficients}=\left[1\,5\,10\right]\right)$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}+5+5x+5y+5z+30z^3+20xy\mathrm{:}6+6x+6y+5z+\mathrm{\dots }\right]$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(xz+y{z}^2+xy{z}^3\,z\right)\:$

$f+z+3$

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

$\mathrm{Add}\left(f\,z+3\right)$

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

$f+\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)$

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

$g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\,\mathrm{PowerSeries}\left(3\right)\right]\,z\right)\:$

$f+g$

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

$h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\,\mathrm{PowerSeries}\left(3\right)\right]\,w\right)\:$

$f+h$

Error, incompatible inputs: expect UnivariatePolynomialOverPowerSeries with the same main variable, but received w <> z

$d≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\&comma;0\&comma;\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\&comma;\mathrm{variables}=\left\{x\&comma;z\right\}\right)$

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

$f+d$

Error, attempted to convert a power series involving z to a univariate polynomial over power series in z, but it is not known to be polynomial in z

$e≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\&comma;0\&comma;\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\&comma;\mathrm{analytic}=z\exp \left(x\right)\right)$

$e≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}z{\&ExponentialE;}^x\mathrm{:}0+\mathrm{\dots }\right]$

$f+e$

$\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\mathrm{+}\left(y\right)z^2\mathrm{+}\left(xy\right)z^3\right]$

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

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