The aim of this chapter is to develop some of the special functions of
analysis, such as the exponential function ex
, and the trigonometric functions, especially
. We shall define these functions and
discover some of their most basic properties (such as the fact that
is periodic, and that it has a zero -- this will be our
definition of the number pi
We will construct all of these functions using infinite series,
so the first part of the chapter is devoted to the basics of
series. Series are a powerful mathematical concept, which are useful for
constructing functions, and have a fascination in their own right. We
shall start by aiming quickly towards the important uses of series, but
as we become more familiar with them, we shall also explore some of the
remarkable features of the more unusual series.
Let (an) be a sequence of real or complex numbers. What does it
mean to talk about the sum of all of these terms:
We can't just say that it is the number we get when we add all of the
terms together. For one thing, that would involve infinitely many
operations, and the process would never be done. Instead we use the idea
of a limit to say that s is the value of the sum if, as we add
more and more of the numbers together, we get closer and closer to
Note that the fact that we are assigning a value to the series includes the
implicit assumption that the series converges, and that this number (zero
in this case) is the limit.
Of course, there's nothing very surprising about the result here! What's important
in the example is not the answer, but how it was obtained. The method of
taking the sum of the first n
terms and forming a sequence of
these sums, then taking the limit of that sequence, is very important.
Notice that we avoided any discussion of the sort: "We're adding zero to
itself infinitely many times, so this is the same as infinity times
zero, and infinity times zero is...?"
The fact that sidesteps this
inconclusive line of speculation is one of the strengths of the
definition of infinite series.
are called the partial sums of the series. A series converges if its
sequence of partial sums converges.
In general, it is impossible to calculate exact expressions for the
partial sums of a series that are any use for determining whether or not
the series converges, or what it converges to. The fact that we've been
able to compute the partial sums of the series in the last three
examples is because they are such simple series. In fact all, the last
three examples fit the model of "geometric series", which are the only
convergent series for which it is possible to compute the partial sums
in an effective manner.
Although it is virtually impossible to calculate explicit formulas for the partial sums of series, nevertheless it is quite easy to determine the convergence or divergence of a huge variety of series. There are a number of "convergence tests" which can be applied to the terms of a series and give effective computational tools for dertermining convergence. Underpinning almost all of these is the following result, the
Comparison Test for Series
. This may well be the single most important result in the convergence of series. (And look out for the roles that the Cauchy sequences, and the completeness of the real numbers, play in this theorem.)
This follows by the comparison between the terms of this series and the series
Note that although it is easy to see that this series converges, by the Comparison Test, it is impossible to find a simple formula for the partial sums of the series. For that matter, it is not at all easy to say what the series converges to.
The contrapositive of the Comparison Test is also useful for recognizing series which do not