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# MATH 2     POWER SERIES:  Radius of Convergence   18 Nov 96
# 
# We look at a couple of series and try to find their radii of
# convergence graphically.
# 
# We first look at the geometric series   1+x+x^2+x^3+...
> Sn := n -> sum(x^k,k=0..n);

                                            n
                                          -----
                                           \     k
                               Sn := n ->   )   x
                                           /
                                          -----
                                          k = 0
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# The sum of this series is   f(x)=1/(1-x)  for -1<x<1
> f := x -> 1/(1-x);

                                             1
                                 f := x -> -----
                                           1 - x
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# Plot the function and some partial sums on the same graph.
# (Specifying a range for y helps the presentation of the
# graph considerably.)
# 
> plot({Sn(1),Sn(2),Sn(4),Sn(10),f(x)},x=-2..2,0..10);
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# We now look at Bessel functions. These arise in the solution
# of differential equations and are used to model real world
# phenomena, such as the temperature distribution in a circular
# plate. These functions are built into Maple --- see the Help
# for more information.
# 
# The partial sums are given by the following formula:
> Tn := n -> sum((-1)^k*x^(2*k)/(2^(2*k)*(k!)^2),k=0..n);

                                       n
                                     -----     k  (2 k)
                                      \    (-1)  x
                          Tn := n ->   )   ------------
                                      /     (2 k)     2
                                     ----- 2      (k!)
                                     k = 0
> 
# Plot some early partial sums and also the Bessel function:
# 
> plot({Tn(1),Tn(2),Tn(4),BesselJ(0,x)},x=-5..5,-1.2..1.2);
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# Plot some higher partial sums on a larger range:
> plot({Tn(8),Tn(20),Tn(50),BesselJ(0,x)},x=-50..50,-0.5..1.2);
>