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+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Series
+ 
+% Address comments and questions to the
+%   Computer Algebra Group, Mathematical Sciences Department
+%   IBM Thomas J. Watson Research Center, Box 218
+%   Yorktown Heights, New York  10598  USA
+ 
+%  Author: Clifton J. Williamson
+%  Date created: 2 November 1989
+%  Date last updated: 2 November 1989
+ 
+\begin{page}{ExSeriesConvert}{Converting Expressions to Series}
+\beginscroll
+You can convert a functional expression to a power series by using the
+function 'series'.
+Here's an example:
+\spadpaste{series(sin(a*x),x = 0)}
+This causes {\em sin(a*x)} to be expanded in powers of {\em (x - 0)}, that is, in powers
+of {\em x}.
+You can have {\em sin(a*x)} expanded in powers of {\em (a - \%pi/4)} by
+issuing the following command:
+\spadpaste{series(sin(a*x),a = \%pi/4)}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesManipulate}{Manipulating Power Series}
+\beginscroll
+Once you have created a power series, you can perform arithmetic operations
+on that series.
+First compute the Taylor expansion of {\em 1/(1-x)}:
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f}}
+Now compute the square of that series:
+\spadpaste{f ** 2 \free{f}}
+It's as easy as 1, 2, 3,...
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesFunctions}{Functions on Power Series}
+\beginscroll
+The usual elementary functions ({\em log}, {\em exp}, trigonometric functions, etc.)
+are defined for power series.
+You can create a power series:
+% Warning: currently there are (interpretor) problems with converting
+% rational functions and polynomials to power series.
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}}
+and then apply these functions to the series:
+\spadpaste{g := log(f) \free{f1} \bound{g}}
+\spadpaste{exp(g) \free{g}}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesSubstitution}{Substituting Numerical Values in Power Series}
+\beginscroll
+Here's a way to obtain numerical approximations of {\em e} from the Taylor series
+expansion of {\em exp(x)}.
+First you create the desired Taylor expansion:
+\spadpaste{f := taylor(exp(x)) \bound{f2}}
+Now you evaluate the series at the value {\em 1.0}:
+% Warning: syntax for evaluating power series may change.
+\spadpaste{eval(f,1.0) \free{f2}}
+You get a sequence of partial sums.
+\endscroll
+\autobuttons\end{page}