Infinite Numerical Series.

To deal with an infinite series, we introduce function notation in Maple. We will use function notation both for the sequence of atoms that "build" the series as well as for the sequence of partial sums that define the series.

>   a :=  n -> 1/n;
>   s :=  n -> sum(a(i), i = 1..n);   # This defines the harmonic series.

We can "sum the series" in two different ways:

>  limit (s(n), n = infinity);   #  Method one.
>  sum (a(n), n = 1.. infinity);    #  Method two.

Of course, we would expect the two methods to yield the same result.
We can also plot the sequence of partial sums as follows:

>  plot (s(n), n = 1..1000);

Notice how slowly the graph rises.  One would probably not guess from looking at the graph that this series diverges!
We might wish to compare the growth of the harmonic sums with that of the log function, since we know that the log function is intimately connected to the area under the curve y = 1/x.

>  plot ({s(n), ln(n)} , n = 1..1000);

What observations do you make from looking at these two graphs?

In the harmonic series example, we already know that the series diverges.  Next let's consider an example of a convergent series.  Here we use the built-in factorial function.

>  a :=  n -> 1/factorial(n);
>   s :=  n -> sum(a(i), i = 1..n);
>  plot (s(n), n = 1..40);
>  limit (s(n), n = infinity);
>  evalf(%);
>  sum (a(n), n = 1.. infinity);
>   evalf(%);

Next, lets try a telescoping series, and see what happens:

>  a :=  n -> 1/ ( n*(n+1) )
>   s :=  n -> sum(a(i), i = 1..n);
>  plot (s(n), n = 1..80);
>  limit (s(n), n = infinity);

Exercises:

1.   Sum the geometric series

In "The Book of Sand" by Jorge Luis Borges , a tall pale stranger comes to the apartment of a book collector and offers to sell a holy book. "He told me his book was called the Book of Sand, because neither the book nor the sand has any beginning or end...The number of pages in this book is no more or less than infinite.   None is the first page, none the last" (Borges 1977).  As the book collector paged through the volume, the stranger said "Look at the illustration closely. You'll never see it again." He realized each page had a different number, typeface, text and language. When he attempted to turn back to a previous page, he was confronted by still more excerpts from unknown works. When he tried to turn to the first or last page of the book, "more pages came between the cover and his thumb."  When the collector realized that he held in his hands an expression of the infinite in a cloth-bound octavo volume, he was profoundly shocked.