The function tpsform(p, var, order) creates a truncated power series (a series data structure) in var.  This truncated series is equal to p up to, but not including, the term containing x^order.

In some cases tpsform can determine when the series is exact and so it drops the $\mathrm{O}\left(x^{\mathrm{order}}\right)$ term. Formal power series created using powpoly fall into this category.

The command with(powseries,tpsform) allows the use of the abbreviated form of this command.

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

$\mathrm{powcreate}\left(t\left(n\right)=\frac{t\left(n-1\right)}{n}\,t\left(0\right)=1\right)\:$

$t\left(\mathrm{\_k}\right)$

$\frac{t\left(\mathrm{\_k}-1\right)}{\mathrm{\_k}}$

$\mathrm{tpsform}\left(t\,x\,6\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+O\left(x^6\right)$