Auxiliary Procedures (execute, but do not change)

>	GS := proc( r, NN )   local N;   if nargs=1 then N:=infinity else N:=NN-1 end if;   return Sum( simplify(r^k,symbolic), k=0..N ) end proc:

>

Lab Overview

A common theme in this course, and the labs, has been the construction of new functions from known functions. This theme can be extended to the construction of Taylor series expansions for functions based on known Taylor series expansions. The power of this method is that you only need to remember a small number of fundamental power series:

          geometric series    =  + ...
                                               =                                                    -1 <  < 1

          exponential            =  + ...
                                          =  

          binomial                = 1 +  +    +    + ...
                                                   =                         -1 <  < 1

where

                           .

The example repeats the derivation of the Taylor series for  presented in class. You will then apply similar techniques to determine power series expansions for other functions. The Lab Questions are very similar to the example, and to exercises in the text.

Example 1: The Power Series for  

Define

>	f := arctan(x);

The key observations are that the derivative of the arctangent function is a rational function

>	Df := diff( f, x );

This provides a definition of the arctangent function as an integral:

>	F := Int( 1/(1+t^2), t=0..x );

To verify that this is correct,

>	F = value( F );

Next, recognize the integrand in this definition as the sum of the geometric series with base . That is, if the geometric series with base  is

>	GS( r ) = value( GS(r) );

Then, the Maclaurin series for  is

>	Df_series := GS( -t^2 );

To verify this, observe that Maple reports

>	Df_series = value(Df_series);

The power series for the arctangent function is obtained by term-by-term integration of the terms of the above series.

>	f_series := Int( Df_series, t=0..x );

It is difficult to get Maple to interchange the integral and the sum and evaluate only the integrals in the above formula. It is much simpler to work with the truncated series with a finite number of terms. For example, the first 10 terms of the power series for  are

>	Df_10 := value( GS( -t^2, 10 ) );

The integration of this polynomial gives the first 10 non-zero terms in the power series for arctangent:

>	F_10 := Int( Df_10, t=0..x );

>	value( F_10 );

>

To confirm this, note that Maple computes the first 10 non-zero terms of the Maclaurin series for the arctangent function to be:

>	series( arctan(x), x=0, 20 );

>

The general form of this power series can be deduced from the above result or by manual integration of the general term in the formula obtained previously:

>

f_series;

>	f_termK := Int( (-1)^k*t^(2*k), t=0..x ): f_termK2 := value(f_termK) assuming k::posint: f_termK =  f_termK2;

>	f_series2 := Sum( f_termK2, k=0..infinity );

To verify this formula, note that the first 10 nonzero terms of this series are

>	series(value(%),x=0,20);

These are in complete agreement with the result found earlier.

>

To address the radius of convergence for the power series for , begin with the fact that the geometric series converges for -1 <  < 1. With , this yields:

   -1 <  < 1

   -1 <  < 1

   -1 <=  < 1

Term-by-term integration preserves this radius of convergence:

   -1 <=  < 1

All that remains is to check the endpoints  and . For :

>	eval( f_series2, x=1 );

Note that the terms of this series alternate sign and  is a decreasing sequence as  -> . Thus, the Alternating Series Test states that this series converges.

For :

>	simplify(eval( f_series2, x=-1 ));

the series is the negative of the series for  and so is convergent. Thus, the interval of convergence for the power series for the arctangent function is [-1,1].

>

Example 2: Integration Example

Question: Evaluate .

Answer: To evaluate this integral, integration by parts with  =  and  (so  = ) leads to

  .

>

Lab Questions