Mathematical Sequences

The aspiring analyst should begin by investigating the folklore of sequences in detail. (Gaughan 33)

Introduction

Sequences play a central role in mathematical analysis and computer science.

Often, mathematicians formally define a sequence as a function whose domain is the positive integers, but go on to say that other index sets, such as the nonnegative integers, are sometimes used. On other occasions, they might refer to any countable set arranged in some order, such as $2, 4, 6, 8, \ldots$ as a sequence. This latter usage certainly seems justified, since countable sets are precisely the kind of images that functions in the first definition require. These two meanings of the term "sequence" can sometimes collide in the form of logical equivocation . In this post, we will examine this conflict, and motivate the definitions. Along the way, we'll examine "index shifts", and finally, compare the results of our analysis with various sources in the literature.

Although the mathematical literature often distinguishes the general or $n\mathrm{th}$ element of a sequence from the general or $n\mathrm{th}$ term of a series, we will ignore this distinction, and use the word "element" and "term" interchangeably and synonymously, whether speaking of sequences or series.

The Initial Index and its Importance

Claim. Let sequences $a$ and $b$ (first definition) and $A$ (second definition) be given such that $A = \left\{ a_n \right\}_{ n=1 }^\infty = \left\{ b_n \right\}_{ n=0 }^\infty.$ Then

1. $a_n$ is the $(n-1)\mathrm{th}$ element of $b$ and the $n\mathrm{th}$ element of $a.$
2. $b_n$ is the $(n+1)\mathrm{th}$ element of $a$ and the $n\mathrm{th}$ element of $b.$
3. $a \neq b,$ yet both generate the same "sequence" (second definition) $A.$ Meaning, the $n\mathrm{th}$ term of a sequence $A$ is not unique, since two different sequences (first definition), $a$ and $b,$ or two different $n\mathrm{th}$ terms $a_n$ and $b_n,$ have the same image $A.$
4. There exist sequences (both definitions) $B = \left\{ b_n \right\}_{ n=1 }^\infty = \left\{ a_n \right\}_{ n=2 }^\infty$ with $A \neq B.$ Meaning, the sequence generated by an arbitrary $n\mathrm{th}$ term, $x_n,$ is not unique, since, for example, $a_n$ generates both $A$ and $B,$ as does $b_n.$

Example. Before we set out to prove this claim, note how the possibility for equivocation, mentioned in the introduction, arose. Had we not called out which definition was intended, we would have seemingly been using the same word in two different ways in the same scope. But are these definitions really different?

To motivate our efforts and by way of example, suppose that $a_n=n$ and $b_n=n+1$ were given. If it were then asked, "Are $a$ and $b$ equal?", what would be your answer? Many would answer "no". But is that true? Certainly $a_n \neq b_n,$ since $n \neq n+1.$ What if further information about $a$ and $b$ were then revealed as follows: $$\begin{equation} \label{eq_n_np1} \left\{ n \right\}_{ n=1 }^\infty = \left\{ n+1 \right\}_{ n=0 }^\infty =1,\ 2,\ 3,\ \ldots,\ n,\ \ n+1,\ \ldots \end{equation}$$ Would your answer change? By equation $\eqref{eq_n_np1}$, $a$ and $b$ now appear to be equal, and yet $a_n$ and $b_n$ are still not equal since $n \neq n+1.$ For that matter, how many sequences are on display in equation $\eqref{eq_n_np1}$? If only one, then $a=b$ should be true, but it is not. Furthermore, what is the $nth$ term of "the" sequence displayed? $n$ or $n+1$? ($a_n$ or $b_n$?)

What's going on here? Let's prove our claim and find out. $$\tag*{$\blacksquare$}$$

Proof. First, $\left\{ b_n \right\}_{ n=0 }^\infty =\left\{ b_{n-1} \right\}_{ n=1 }^\infty =\left\{ a_n \right\}_{ n=1 }^\infty,$ so $a_n=b_{n-1}$ and $b_n=a_{n+1},$ proving (i) and (ii). Next, note that $a$ and $b$ have the same image $A,$ but different domains, so they are not equal functions, proving (iii). Finally, select the subsequence of $A$ that results from "deleting" the first element $a_1=b_0,$ leaving us with the new sequence $B = \left\{ b_n \right\}_{ n=1 }^\infty = \left\{ a_n \right\}_{ n=2 }^\infty,$ thus proving (iv). $$\tag*{$\blacksquare$}$$

Example. Let $A$ be the arithmetic progression (sequence) whose first element is $a_1=b_0=23$ with common difference $d=-6.$ If indexing starts at $n=1,$ then the general element of this sequence is $a_n=29-6n.$ If indexing starts at $0,$ the general element is $b_n=23-6n.$ Thus, we have $A = \left\{a_n\right\}_{n=1}^\infty = \left\{29-6n\right\}_{n=1}^\infty = \left\{23-6n\right\}_{n=0}^\infty = \left\{b_n\right\}_{n=0}^\infty.$ The first few elements of $A$ are given below: $$\begin{align*} &a_1 = b_0 = 29-6 = 23 \\ &a_2 = b_1 = 29-12 = 17 \\ &a_3 = b_2 = 29-18 = 11 \\ &a_4 = b_3 = 29-24 = 5 \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ &a_n = b_{n-1} = 29-6n \\ &a_{n+1} = b_n = 23-6n \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \end{align*}$$

This example illustrates our claim as follows: $a_n=b_{n-1}=29-6n$ is the $n\mathrm{th}$ element of $a$ and the $(n-1)th$ element of $b,$ and $b_n=a_{n+1}=23-6n$ is the $n\mathrm{th}$ element of $b$ and the $(n+1)th$ element of $a,$ illustrating (i) and (ii).

Furthermore, two different $n\mathrm{th}$ terms, $29-6n$ and $23-6n,$ generate the same sequence $A=23,\ 17,\ 11,\ 5,\ \ldots,\ 29-6n,\ 23-6n,\ \ldots,$ because $a$ and $b$ have different domains, illustrating (iii).

Finally, if we shift $b$’s domain to be the same as $a$ (the positive integers), we obtain a new function $b',$ defined by $b_n'=b_n$ for all $n>0,$ that no longer generates $A$ but a new sequence $B = \left\{b_n\right\}_{n=1}^\infty = \left\{23-6n\right\}_{n=1}^\infty = 17,\ 11,\ 5,\ldots,\ 29-6n,\ 23-6n,\ \ldots,$ which is the subsequence of $A$ obtained by "deleting" the first element of $A,$ that is $a_1=b_0=23.$ Also, notice the appearance of $a_n$ in $B$ as the $\left(n-1\right)th$ element of $b'.$ However, since $a_1=23\notin B,$ we realize we have a new function $a',$ defined by $a_n'=a_n$ for all $n>1,$ that also generates the same sequence $B.$ Therefore, $a_n$ is the $n\mathrm{th}$ term of $a'$ and $b_n$ the $n\mathrm{th}$ term of $b',$ and we see that this shift of index applied to $a$ and $b$ obtained the new sequence $B=\left\{b_n\right\}_{n=1}^\infty=\left\{a_n\right\}_{n=2}^\infty,$ with $A\neq B,$ illustrating (iv): an arbitrary $n\mathrm{th}$ term, $x_n,$ generates countably infinite different sequences, one for each possible index shift or domain. $$\tag*{$\blacksquare$}$$

Uses in Computer Science

In many significant programming languages, such as C and Java, arrays are indexed by default with a range of integers starting at $0,$ offering an analogy to finite sequences in mathematics. In such languages, the element in the $n\mathrm{th}$ position of an array, $a,$ could typically be accessed with a syntax such as $a[n],$ in analogy to the mathematical $n\mathrm{th}$ term $a_n.$ The element of the array at the $n\mathrm{th}$ index is understood to be in the $(n+1)th$ position of the array, and the programmer who fails to understand this will write code that leads to "index out of range" errors.

At first, this treatment may seem different than the use in mathematics, but it is not. In both contexts, indexing the initial element at $0$ leads to the logical equivocation that the $0th$ (zeroth) term of the sequence/array is the $1st$ (first) term, the first term is the second, $\ldots,$ the $n\mathrm{th}$ term the $(n+1)th$ term, and so on. It is as though you, the reader, are expected to apply two different definitions to the same word in the same sentence simultaneously. In fact, you are! If that bothers you, let’s try to convince ourselves.

That the equivocation is real is proved by our claim and example given earlier: Given a sequence $A = \left\{a_n\right\}_{n=1}^\infty = \left\{b_n\right\}_{n=k}^\infty,\ k\neq1,$ the initial element $b_k$ is the "first" element of $a$ but the $kth$ element of $b,$ yet $a$ and $b$ form the same sequence $A = a_1,\ a_2,\ \ldots,\ a_n,\ \ldots = b_k,\ b_{k+1},\ \ldots,\ b_n,\ \ldots.$ But let’s approach it a different way.

In the mathematical literature, as in computer science, the $n\mathrm{th}$ element always unambiguously means the element at index $n$; but then, it must follow, given a sequence $x_0,x_1,\ x_2,\ \ldots.,$ and substituting $1$ for $n$ in the phrase "$n\mathrm{th}$ term”, that $x_1$ is the "first term", and $x_0$ is the "zeroth term". That follows from simple substitution on "$n\mathrm{th}$ term”. Even though $x_0$ is clearly the first term in the usual sense! Comparing with our formal analysis, we can see that our two senses of ordinal position correspond to the two functions $a$ and $b,$ with different domains, used to index the one sequence $A,$ and each function has its own $n\mathrm{th}$ term, $a_n$ and $b_n,$ corresponding to our two senses of $n\mathrm{th}$ term for sequence $A.$ Formally, all such difficulties are avoided under the traditional definition of sequence, that prescribes a single index set for all sequences, an index set that coincides with the traditional sense of "first", the positive integers. Informally, common practice allows multiple alternative definitions, one for each possible shift of index, especially one based on $0$ as the first element, and the equivocation arises.

"Equivocation" puts a name to the "abuse of language" that is not without precedent in the mathematical literature, as we shall see in the next section.

Usage in the Mathematical Literature

Let's grade the motivation I have given for the traditional definitions and usage of "sequence", "$n\mathrm{th}$ term", and "index shifts", by comparing these from different well-known sources in the mathematical literature. (Gaughan 33) writes,

A sequence is a function whose domain is the set of positive integers. If $a$ is a sequence, it is customary to write $a\left(n\right)=a_n$ for each positive integer $n$ and write $a=\left\{a_n\right\}_{n=1}^\infty.$ We call $a_n$ the $n\mathrm{th}$ term of the sequence.

A quick search in the same text does not find a definition or usage of "general element" or "general term," suggesting that "general element" may merely be a synonym or alternative name, used by some authors, to refer to the "$n\mathrm{th}$" element when the initial index $n=1$ is used.

When speaking of series, (Gaughan 173) writes,

It is often convenient to index the terms of an infinite series beginning with an integer other than $1.$ As our discussion unfolds, it will be clear that questions of convergence are independent of whether we index the terms beginning with $n=1$ or $n=p$ for some other integer $p.$

Subsequently, (Gaughan 176) offers Example 6.4,

The series $\sum_{n=0}^{\infty}r^n$ is called the geometric series with ratio $r.$

Notably, however, the author does not deviate from using initial index $n=1$ when using sequences to show the intervals of convergence for this series in the same example. This seems deliberate, since, as already mentioned, the author explicitly defined a sequence as a function whose domain is the natural numbers.

In contrast, (Finney, Weir and Giordano 608) offer the following definitions:

An infinite sequence of numbers is a function whose domain is the set of integers greater than or equal to some integer $n_0.$ $\ldots$ Usually, $n_0$ is $1$ and the domain of the sequence is the set of positive integers. Sometimes, however, we want to start sequences elsewhere. We take $n_0=0$ when we begin Newton’s method. We might take $n_0=3$ if we were defining a sequence of $n-sided$ polygons. $\ldots$ The number $a(n)$ is the $n\mathrm{th}$ term of the sequence, or the term with index $n.$

Here, the authors seem to treat $a_n$ as the $n\mathrm{th}$ term even when the initial index is $n\neq1,$ suggesting the abuse is merely a convention, one the reader should get comfortable with. Similar abuses occur elsewhere in mathematics. For example, (Gaughan 174) writes,

Perhaps a word of apology should be offered for using $\sum_{n=1}^{\infty}a_n$ as a name both for an infinite series and for the real number that is the limit of the sequence of partial sums when the series converges. However, this abuse conforms to convention and the reader’s experiences. Note that, if the series does not converge, we do not use $\sum_{n=1}^{\infty}a_n$ to denote a real number.

About geometric series, (Finney, Weir and Giordano 630) write,

Geometric series are series of the form $a+ar+ar^2+\cdots+ar^{n-1}+\cdots=\sum_{n=1}^{\infty}{ar^{n-1}}$

thus opting for an initial index of $1,$ in contrast to the corresponding treatment mentioned earlier by Gaughan. Interestingly, however, the authors point out that

The equation $\sum_{n=1}^{\infty}{ar^{n-1}} = \frac{a}{1-r},\ \left|r\right|<1,$ holds only if the summation begins with $n=1.$

Turning to (Stevens 520-522), we find explicit use and definition of the phrase "general element" as a synonym for the $n\mathrm{th}$ element, but initial index $n=1$ is imposed:

A function $f,$ whose domain is the set of all positive integers, is an infinite sequence function. The elements in the range of $f,$ taken in the order $f\left(1\right),\ f\left(2\right),\ f\left(3\right),\ldots,$ form an infinite sequence. If the domain of a sequence function is not stated, we assume it to be the set of all positive integers $\left\{1,\ 2,\ 3,\ \ldots\right\}.$ $\ldots$ It is customary to denote the first element of a sequence as $a_1,$ the second element as $a_2,$ the third element as $a_3,$ and so on. The $n\mathrm{th}$ element of a sequence, denoted as $a_n,$ is called the general element of the sequence.

Finally, (Rudin 26) writes:

By a sequence, we mean a function $f$ defined on the set $J$ of all positive integers. If $f\left(n\right)=x_n,$ for $n\in J,$ it is customary to denote the sequence $f$ by the symbol $\left\{x_n\right\}$ or sometimes by $x_1,\ x_2,\ x_3,\ \ldots.$ The values of $f,$ that is, the elements $x_n,$ are called the terms of the sequence. If $A$ is a set and if $x_n\in A$ for all $n\in J,$ then $\left\{x_n\right\}$ is said to be a sequence in $A,$ or a sequence of elements of $A.$ Note that the terms $x_1,\ x_2,\ x_3,\ \ldots$ of a sequence need not be distinct. Since every countable set is the range of a $1-1$ function defined on $J,$ we may regard every countable set as the range of a sequence of distinct terms. Speaking more loosely, we may say that the elements of any countable set can be “arranged in a sequence”. Sometimes it is convenient to replace $J$ in this definition by the set of all nonnegative integers, i.e., to start with $0$ rather than with $1.$

In Rudin’s last paragraph above, we see the most explicit statement that an index shift amounts to a new definition.

Conclusion

In this post, I have tried to motivate the most common definitions of a sequence and it’s $n\mathrm{th}$ term, as well as reconcile the logical equivocation that results when using "both" definitions in the same context, such as when shifting indices, or when exploring a relationship between two sequences that use different index sets, common practices in the literature.

Works Cited

Finney, Ross L., et al. Thomas' Calculus. Tenth Edition. Boston: Addison Wesley Longman, 2001. Print.

Gaughan, Edward D. Introduction to Analysis. Fifth Edition. Providence: American Mathematical Society, 1998. Print.

Rudin, Walter. Principles of Mathematical Analysis. Third Edition. McGraw Hill, 1976. Print.

Stevens, David E. College Algebra. St. Paul: West Publishing Co., 1994. Print.