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# MATH 2     ALTERNATING SERIES    11 November 1996
# 
# We look at a couple of examples of alternating series.
# 
# The first series is  1 - 1/2 + 1/3 - 1/4 + ...
# 
> a := k -> (-1)**(k-1) / k;

                                            (k - 1)
                                        (-1)
                              a := k -> -----------
                                             k
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# We set  Sn to be the partial sums:
> Sn := n -> sum(evalf(a(k)), k=1..n);

                                       n
                                     -----
                                      \
                          Sn := n ->   )   evalf(a(k))
                                      /
                                     -----
                                     k = 1
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# Calculate some of the partial sums for even n:
> Sn(4); Sn(8);

                                   .5833333333

                                   .6345238095
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# ... and for some of the odd n:
> Sn(5); Sn(7);

                                   .7833333333

                                   .7595238095
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> 
# Plot the partial sums:
> P := [[n,Sn(n)] $n=2..100]: plot(P,style=point);
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> 
# Find Sn for some large values of n:
> Sn(101); Sn(infinity);

                                   .6980731694

                                   .6931471806
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# This limit can actually be written in simple form
# (see section 10.10):
> evalf(ln(2));

                                   .6931471806
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# Maple actually has a formula for the partial sums,
# using something called the Psi function:
> Sn(n);

                          n  n + 1.      n + 1.
    .6931471806 - 1. (-1.)  (------ - 1. ------ + .5000000000
                                n        n - 1.

        (Psi(.5000000000 n) - 1. Psi(.5000000000 n - .5000000000)) (n + 1.))/

        (n + 1.)
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# Our second example is
# 1 - x + (x**2)/2! - (x**3)/3! + ...
# 
> a2 := k -> (-1)**(k-1) * x**(k-1) / (k-1)!;

                                        (k - 1)  (k - 1)
                                    (-1)        x
                         a2 := k -> --------------------
                                          (k - 1)!
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# We set a2(1)=1, since 0! is defined to be 1:
> a2(1) := 1;

                                   a2(1) := 1
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# Let b(k) = (x**(k-1))/(k-1)!
# We get b(k) -> 0 as k -> infinity.
# Also, b(k+1) < b(k) provided k > x.
# So this series is convergent for any x.
# 
# Partial sums:
> Tn := n -> a2(1) + sum(a2(k), k=2..n);

                                           /  n        \
                                           |-----      |
                                           | \         |
                        Tn := n -> a2(1) + |  )   a2(k)|
                                           | /         |
                                           |-----      |
                                           \k = 2      /
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# Eg, take x=1.5 and evaluate some of the partial sums:
> x := 1.5;

                                    x := 1.5
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> Tn(4); Tn(9); Tn(16);

                                   .0625000000

                                   .2232221331

                                   .2231301601
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> Tn(101); Tn(infinity);

                                   .2231301601

                                   .2231301601
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# We plot the partial sums.
# Two points:
#   (1)  We define Q explicitly as a sequence ---
#        Maple didn't like it when we used a format similar
#        to the definition of P.
#   (2)  We start the partial sums at n=2 because Tn(1) requires
#        taking a sum from k=2 to k=1 (ie, an empty sum), and
#        the definition of this can be confusing to Maple.
> Q:=[seq([n,Tn(n)], n=2..20)]: plot(Q,style=point);
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> evalf(exp(-1.5));

                                   .2231301601
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