Example 3.1.1.
Consider the constant sequence given by the formula \(a_n=1\text{.}\) Then making the list we find
\begin{equation*}
(a_n)=(1,1,1,1,1,1\ldots)
\end{equation*}
while the range is simply the single element set \(\{1\}\text{.}\)
Section 3.1 Definition and notation
A sequence is a function whose inputs are the positive integers, and whose outputs are real numbers. So in the notation of functions, a sequence is a function \(f: {\mathbb N} \longrightarrow {\mathbb R}\text{.}\)
But just like the notation \(f(x)\) is the traditional one and most commonly used for the output of functions studied in Calculus (where \(x\) is a real number), sequences have their own notation. So the input is usually denoted by \(n\text{,}\) and instead of using parentheses for the input we use a subscript. Also, while the most common letters to denote the names of functions are \(f,g,h\text{,}\) for sequences we will most commonly use \(a,b,c\text{,}\) sometimes \(x,y,z\text{.}\)
So if we use \(a\) as the name of the function, the output corresponding to input \(n\) is \(a_n\text{.}\) Sometimes it is convenient to include \(0\) in the domain, so that the domain of the sequence is the set on non-negative integers.
We use the notation \((a_n)\) to describe the whole sequence, rather than an individual output value \(a_n\text{.}\) Sometimes, to explicitly show the domain, we write \((a_n)_{n=1}^\infty\text{,}\) or \((a_n)_{n=0}^\infty\text{.}\) The most important feature of a sequence is that its output values can be listed:
\begin{equation*}
(a_n)_{n=1}^\infty = (a_1,a_2,a_3, \ldots )
\end{equation*}
or
\begin{equation*}
(a_n)_{n=0}^\infty = (a_0,a_1,a_2, \ldots ).
\end{equation*}
Subsection 3.1.1 Examples
It is important to distinguish a sequence from its range. A sequence (being a special type of function) must describe what output will correspond to each input. Making a list \((a_1,a_2,\cdots)\) will surely accomplish that. But the range carries a lot less information: it is just the set of all the outputs of the sequence. So the range of \((a_n)\) is the set \(\{a_n: n\geq 1\}\text{.}\) The next example shows that the difference can be dramatic.
The fact that sequences are lists of numbers often makes it possible for us to “guess” what a formula for it might be. We now discuss the most common and important examples.
Example 3.1.2.
\((a_n)=(1,2,3,4,\ldots)\text{.}\) It seems evident that this is the sequence that lists all the natural numbers in increasing order, starting from \(1\text{,}\) and so \(a_n=n\text{.}\) BUT, it’s important to realize that if we are only given the first few values (in this case only the first four), our guess is nothing but a guess. How do we know that from the fifth entry on the sequence is not what we think it is? Maybe if could look further we would find something like \((a_n)=(1,2,3,4,3,2,1,2,3,4,3,2,1,\ldots)\text{,}\) so that our guess was quite wrong. So we must always be careful to draw conclusions from looking at just a few values. In spite of this warning, it is often a very good idea to try to guess what a sequence might be, because it will give us a hint of what we may be able to prove from our initial guess.
Example 3.1.3.
\((a_n)=(2,4,6,8,\ldots)\text{.}\) Evidently these look like the even numbers, and our guess for a formula is \(a_n=2n\text{.}\)
Example 3.1.4.
\((a_n)=(1,3,5,7,\ldots)\text{.}\) These must be the odd numbers, \(a_n=2n-1\text{.}\)
Example 3.1.5.
\((a_n)=(1,4,9,16,\ldots)\text{.}\) It definitely looks like we are dealing with the squares: \(a_n=n^2\text{.}\)
Example 3.1.6.
\(\displaystyle (a_n)=\left(\frac{1}{1}, \frac{1}{2},\frac{1}{3},\ldots\right)\text{.}\) Our guess of course is that these are the reciprocal numbers, \(\displaystyle a_n=\frac{1}{n}\text{.}\)
Example 3.1.7.
\((a_n)=(-1,1,-1,1,-1,1\ldots)\text{.}\) This looks like the alternating sequence \((-1)^n\text{.}\)
The alternating sequence \(a_n=(-1)^n\) is the most basic example of one that alternates between two values (in this case \(-1\) and \(1\)). Simple modifications allow us to find a formula for sequences that alternate between two given numbers (see the exercises).
Subsection 3.1.2 Properties of sequences
We have already mentioned that the range \(\{a_n: n\in {\mathbb N}\}\) of a sequence is a subset of \(\mathbb R\text{.}\)
Definition 3.1.8.
We say that a sequence \((a_n)\) is bounded if its range is a bounded subset of \(\mathbb R\text{.}\) So a sequence is bounded if there is some \(M\in {\mathbb R}\) such that \(|a_n|\leq M\) for all \(n\text{.}\)
Remark 3.1.9.
This is an example of terminology that uses the same adjective (bounded) for two different nouns (mathematical objects in this case): a bounded set is a set, and a bounded sequence is a sequence. In ordinary English language, if we explain to a child what “large” in “large house” means, we probably do not need to explain what “large” in “large dog” is. But in mathematics, if we use an adjective applied to a noun, we need to always make sure that we have defined its meaning for that particular noun. So having defined “bounded subset” does not at all explain what “bounded sequence” means.
Definition 3.1.10.
We say that a sequence \((a_n)\) is increasing if \(a_{n+1}\geq a_n\) for all \(n\text{,}\) and it is decreasing if \(a_{n+1}\leq a_n\) for all \(n\text{.}\) If \(a_{n+1}\gt a_n\text{,}\) we say it is strictly increasing and if \(a_{n+1}\lt a_n\) it is strictly decreasing. We say that \((a_n)\) is monotone if it is either increasing or decreasing, and strictly monotone if it is either strictly increasing or strictly decreasing.
Example 3.1.11.
The sequence \(a_n=n\) is strictly increasing, and \(a_n=\frac{1}{n} \) is strictly decreasing. Any constant sequence \(a_n=c\) is both increasing and decreasing. For a more interesting example, consider the sequence \(a_n\) defined by \(a_n=\)the number of perfect squares no larger than \(n\text{,}\) or in symbols \(a_n=|\{m\in {\mathbb N} : m^2\leq n\}|\text{.}\) So for example \(a_1=a_2=a_3=1\text{,}\) \(a_4=a_5=a_6=a_7=a_8=2\text{,}\) \(a_9=3\) and so on. Then \((a_n)\) is increasing. The simplest example of a sequence that is not monotone is the alternating sequence \(a_n=(-1)^n\text{.}\)
