Series

Maple knows how to generate any finite number of terms of either a
Taylor (Laurent) series or an asymptotic series expansion of some
expression.  The Maple functions are taylor and asympt respectively. 
You specify the number of terms by specifying the "order of truncation".
 If the terms up to the x^(n-1) do not make the complete expression, and
order term of O(x^n) is added to the end.  In the case of the Taylor
series, you can specify the point of expansion for the series.

> # Compute some terms of the Taylor series for cos(z) at z=0:
> taylor(cos(z),z=0);
         2         4      6
1 - 1/2 z  + 1/24 z  + O(z )
> # The default number of terms in the expansion is controlled by the
> # global variable Order, which is initially set to 6:
> Order := 9;
Order := 9
> taylor(cos(z),z=0); 
         2         4          6            8      9
1 - 1/2 z  + 1/24 z  - 1/720 z  + 1/40320 z  + O(z )

> # You can also specify the expansion order to taylor(..):
> taylor(cos(z),z=0,4);  
         2      4
1 - 1/2 z  + O(z )

There is also a series function for a more general method of series
expansion.  See help('series'), help('taylor'), and help('asympt') for
more information on series-expansion facilities.