% Copyright The Numerical Algorithms Group Limited 1991.
% 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}
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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)}
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\begin{page}{ExSeriesManipulate}{Manipulating Power Series}
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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,...
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\begin{page}{ExSeriesFunctions}{Functions on Power Series}
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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 (interpreter) 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}}
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\begin{page}{ExSeriesSubstitution}{Substituting Numerical Values in Power Series}
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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.
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