4.3 Sequences and Series: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Given an ordered infinite list of mathematical objects (numbers, functions, sets, etc.), we denote the sequence
\[a_1, a_2, a_3, \dots .\]
This is read out as a one, a two, a three dot dot dot. Alternatively, one can also read this as a sub one, a sub two, a sub three dot dot dot.
Note here that the word sub is an abbreviation of the word subscript, so [latex]a_1[/latex] is a subscript one. Furthermore, note that dot dot dot refers to the ellipsis symbol [latex]\dots[/latex], but alternatively, one can instead say and so on.
The objects that make up the sequence are called the terms of the sequence. So, in the above case, [latex]a_1[/latex] is the first term of the sequence, [latex]a_2[/latex] is the second term of the sequence, and so on. Each term is identified by its position number, called the index. Note that it is also common to start indexing sequences from 0. That is,
\[a_0, a_1, a_2, a_3, \dots ,\]
where [latex]a_0[/latex] is often called the zero-th term.
Sequences are usually expressed in terms of their general term, [latex]a_n[/latex],
[latex](a_n)_n[/latex] or just simply [latex](a_n)[/latex],
both of which are read out as the sequence a n.
The general term [latex]a_n[/latex], is also often called the [latex]n[/latex]-th term. For more details about the term [latex]n[/latex]-th, see the Language Focus in Section 2.3.
If we want to make it explicit that we are dealing with an infinite sequence, we can write
\[(a_n)_{n=1}^{\infty} \]
which is read out as the sequence a n where n goes/runs from one to infinity.
For example, if the general term is [latex]\frac{1}{2^n}[/latex], then the sequence is
\[\left(\frac{1}{2^n}\right)_{n=1}^{\infty} = \frac{1}{2},\frac{1}{4},\frac{1}{8}, \dots. \]
This is read out as the sequence one over two to the n where n goes from one to infinity is equal to, one over two, one over four, one over eight, and so on.
If a sequence converges to a finite limit, we write
\[a_n \to a \text{ as } n \to \infty, \quad \text{ or } \quad \lim_{n \to \infty} a_n =a, \]
where [latex]a[/latex] is called the limit of the sequence [latex](a_n)[/latex]. These can be read out as a n tends to/converges to/approaches a as n tends to infinity and, the limit of a n as n tends to infinity is a, respectively. The second expression, can also be read as the limit as n tends to infinity of a n is a.
Furthermore, note that when we get closer and closer to a finite number a, we can say tends to a, or converges to a, or approaches a. But when we are talking about getting closer to infinity, we can say tends to infinity, or approaches infinity, but not converges to infinity. This is because in mathematical language convergence means approaching a finite limit.
The property is called convergence and the adjective is convergent (e.g. a convergent sequence). A sequence that does not converge is said to diverge, and similarly, the property is called divergence and the adjective divergent (e.g. a divergent sequence).
Language Focus
It is worth noting how the general and the mathematical meaning of convergence relate to each other. In everyday language, we say that two roads converge when they come together and meet at a point, or that people converge when they gather from different locations. In mathematics, convergence has a similar meaning: when a sequence converges, its terms move closer and closer to a specific value, called the limit. However, there is an important difference. In everyday language, convergence usually means that things actually meet or arrive. In mathematics, convergence means approaching a limit without necessarily reaching it. For example, the sequence [latex]1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots[/latex] converges to [latex]0[/latex], but the terms never actually equal [latex]0[/latex].
A sequence where the difference between any two consecutive terms is constant, is called an arithmetic sequence. This constant difference is called the common difference, and it is usually denoted by [latex]d[/latex] (d for difference). An example of an arithmetic sequence is [latex]3, \, 8, \, 13, \, 18, \dots[/latex], with common difference [latex]d=5[/latex].
A sequence in which the ratio between any two consecutive terms is constant, is called a geometric sequence. This constant ratio is called the common ratio, and it is usually denoted by [latex]r[/latex] (r for ratio). An example of a geometric sequence is [latex]3, \, 15, \, 75, \, 375, \dots[/latex], with common ratio [latex]r=5[/latex].
Mathematics Focus
To understand why these sequences have their particular names, we first need to introduce the following concepts.
Given two real numbers [latex]a[/latex] and [latex]b[/latex],
- their arithmetic mean is defined as [latex]\frac{a+b}{2}[/latex],
- and their geometric mean is defined as [latex]\sqrt{ab}[/latex].
Observe that in an arithmetic sequence, any given term (except the first) is the arithmetic mean of the term before it and the term after it. Similarly, in a geometric sequence with positive terms, any given term (except the first) is the geometric mean of the term before it and the term after it. These statements are straightforward to show, and we encourage you to prove them.
A sequence in which the terms alternate between positive and negative values, is called an alternating sequence. An example of an alternating sequence is any geometric sequence where the common ratio is negative. For example, [latex]3, \, -15, \, 75, \, -375, \dots[/latex], where the common ratio [latex]r=-5[/latex].
Given a sequence [latex](a_n)_n[/latex], we denote the sum of the terms of the sequence to be the series,
\[\sum_{n=1}^{\infty} a_n =a_1 +a_2+a_3+ \cdots .\]
This is read out as the sum from n equals one to infinity of a n is equal to, a one, plus a two, plus a three, plus dot dot dot. Note that the left-hand side can also be read out as the sum of a n from n equals one to infinity.
As before, [latex]a_n[/latex] is called the general term of the series.
Language Focus
Note that the word series is an irregular noun, as it has the same singular and plural form. We say here is one series but also here are two series.
Additionally, note that the mathematical meaning of the word sequence is related to its general meaning, which refers to a particular order in which related things follow each other. For example, What is the sequence of events? However, we should be careful not to confuse the mathematical meaning of the word series with its general meaning. The general meaning of series is a number of related things that occur one after another. For example, a TV series/book series consists of a multiple episodes/books that follow a continuous storyline and come one after another.
The sum of the first [latex]N[/latex] terms of a series is known as the partial sum, and it is denoted by
\[\sum_{n=1}^{N} a_n =a_1 +a_2+a_3+ \cdots + a_N.\]
This is read out as the sum from n equals one to capital n of a n is equal to, a one, plus a two, plus a three, plus dot dot dot, plus a capital n.
Mathematics Focus
Partial sums, usually denoted by [latex]S_N=\sum_{n=1}^{N} a_n[/latex], allow us to determine whether an infinite series converges or diverges. Indeed, an infinite series is formally defined as the limit of the sequence of its partial sums: we say the series [latex]\sum_{n=1}^{\infty} a_n[/latex] converges to a value if and only if [latex]\lim_{n \to \infty} S_n =S[/latex].
Note that the summation symbol [latex]\sum[/latex] is the capital Greek letter sigma, which is the equivalent of the English letter S, and it stands for summation, reminding us that we are adding up all the terms of the sequence. Just as the integral sign [latex]\int[/latex] is an elongated S that represents continuous summation (adding infinitely many infinitesimal strips), the sigma symbol represents discrete summation (adding infinitely many individual terms in a sequence). See Section 3.3 for more details.
An arithmetic series is simply the sum of the terms of an arithmetic sequence. And similarly, a geometric series is the sum of the terms of a geometric sequence.
The harmonic series is a specific series whose terms are the reciprocals of the positive integers
\[\sum_{n=1}^{\infty}\frac{1}{n} =1+ \frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots.\]
This is read out as the sum from n equals one to infinity of one over n is equal to, one, plus a half, plus a third, plus a quarter, plus a fifth, plus dot dot dot.
The harmonic series is named after the concept of harmonics in music, where the frequencies of musical overtones are related by integer ratios.
Mathematics Focus
Mathematically, the harmonic series is particularly interesting because it appears to converge at first glance, since the terms [latex]\frac{1}{n}[/latex] approach zero as [latex]n[/latex] increases. However, the harmonic series actually diverges, since the sequence of partial sums [latex]S_N = 1 +\frac{1}{2}+\frac{1}{3} + \cdots + \frac{1}{N}[/latex] grows to infinity, even though very slowly.
This makes the harmonic series a classic counterexample in mathematics when showing that the terms of a series approaching zero is a necessary, but not sufficient condition for the series to converge. In other words, although the terms must approach zero for a series to converge, the fact that the terms approach zero does not guarantee that the series converges.
So far, we have seen series with constant terms. However, some series are defined in terms of a variable, where each term is a function of [latex]x[/latex]. This means the series gives us different values, depending on what value we substitute for [latex]x[/latex].
A power series is a series of the form
\[ \sum_{n=0}^{\infty} c_n (x - a)^n,\]
where [latex](c_n)[/latex] is a sequence of constants and [latex]a[/latex] is also a constant.
This is read out as the sum from n equals zero to infinity of c n times, x minus a, to the power n.
It is called a power series because each term in the series involves a power of [latex](x-a)[/latex], where the power increases with each successive term.
A Taylor series is a specific type of power series in which the coefficients are determined by the derivatives of a function. The Taylor series of a function [latex]f(x)[/latex], expanded about the point [latex]a[/latex] is denoted by
[latex]\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.[/latex]
This is read out as the sum from n equals zero to infinity of, the nth derivative of f evaluated at a, divided by n factorial, times x minus a, to the n.
Mathematics Focus
Taylor series are named after the mathematician Brook Taylor, who developed this method in the early 18th century. A Taylor series expanded about [latex]a=0[/latex] is called a Maclaurin Series, named after the mathematician Colin Maclaurin. A Taylor series allows us to represent a function as an infinite sum of polynomial terms, which are much simpler to work with. This is particularly useful when we need to approximate functions or analyse their behaviour near a specific point.
Given a function [latex]f(x)[/latex], its Taylor Series does not always converge, and even when it does converge, it may not converge to the original function [latex]f(x)[/latex] for all values of [latex]x[/latex]. Consider for example, [latex]f(x)= e^{-1/{x^2}}[/latex] for [latex]x \neq 0[/latex], and [latex]f(0)=0[/latex].
The interval of convergence is the set of real [latex]x[/latex]-values for which the power series converges. For a Taylor series, one must separately check whether the sum of the series equals the original function on that interval. Analogously, in the complex plane (see Chapter 5), the set where a power series converges is typically a disk centred at the expansion point. The disk is called the disk of convergence its radius is called the radius of convergence.
This shows that writing down an infinite series is one thing, but determining whether and where it converges, is quite another issue.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
