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Chapter 6 Sequences and Series

In this Chapter, we introduce the concepts of infinite sequences and series. One important application is the representation of a differentiable function as an infinite sum of powers of the independent variable. This allows us to extend differentiation and integration to a more general class of functions that cannot necessarily be represented in terms of elementary functions like power, trigonomentric, exponential and logarithmic functions.

Consider the following sum:

\begin{equation*} {1\over2}+{1\over4}+{1\over8}+{1\over16}+\cdots+{1\over2^i}+\cdots \end{equation*}

The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite sum? While at first it may seem difficult or impossible, we have certainly done something similar when we talked about one quantity getting “closer and closer” to a fixed quantity. Here we could ask whether, as we add more and more terms, the sum gets closer and closer to some fixed value. That is, look at

\begin{align*} \frac{1}{2}\amp =\frac{1}{2}\\ \frac{3}{4}\amp =\frac{1}{2}+\frac{1}{4}\\ \frac{7}{8}\amp =\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\\ \frac{15}{16}\amp =\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16} \end{align*}

and so on, and consider whether these values have a limit. It seems likely that they do, namely \(1\text{.}\) In fact, as we will see, it's not hard to show that

\begin{equation*} {1\over2}+{1\over4}+{1\over8}+{1\over16}+\cdots+\frac{1}{2^i} = {2^i-1\over2^i}=1-{1\over2^i} \end{equation*}

and then

\begin{equation*} \lim_{i\to\infty} \left(1-\frac{1}{2^i}\right) = 1 - 0 = 1\text{.} \end{equation*}

There is a context in which we already implicitly accept this notion of infinite sum without really thinking of it as a sum: The representation of a real number as an infinite decimal. For example,

\begin{equation*} 0.3333\bar3 = {3\over10}+{3\over100}+{3\over1000}+{3\over10000}+\cdots= {1\over3}\text{,} \end{equation*}

or likewise

\begin{equation*} 3.14159\ldots = 3+{1\over10}+{4\over100}+{1\over1000}+{5\over10000}+ {9\over100000}+\cdots = \pi\text{.} \end{equation*}

An infinite sum is called a series, and is usually written using the same sigma notation that we encountered in Chapter 1. In this case, however, we use infinity as an upper limit of summation to indicate that there is no ‘last term’. The series we first examined can be written as

\begin{equation*} \frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^i} + \cdots=\sum_{i=1}^{\infty}\frac{1}{2^i} \end{equation*}

A related notion that will aid our investigations is that of a sequence. A sequence is just an ordered (possibly infinite) list of numbers. For example, the terms in the infinite sequence above are an example of a sequence:

\begin{equation*} 1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots \end{equation*}

We will begin by learning some useful facts about sequences.