Describing Sequences - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Describing Sequences
Describing Functions

A sequence may be thought of as a function defined over the natural numbers, or, more generally, over a subset of the integers (Sect. 3.1). Indeed in number theory the real (or complex) sequences are called arithmetical functions. Using this analogy, some terminology introduced for real functions translates literally to real sequences. So the terms

positive, negative, increasing, decreasing, monotonic,constant, periodic, bounded, bounded away from zero

have the same meaning for sequences as they have for functions. We write

The sequence of primes is positive, increasing, and unbounded.

Other terms require amendments or are simply not relevant to sequences. For instance, injectivity is not used—we just say that the terms of a sequence are distinct—while surjectivity is rarely significant. Invertibility is used in a different sense than for functions—see Sect. 9.4. The terms ’even’ and ’odd’ are still applicable to doubly-infinite sequences.

A sequence which settles down to a periodic pattern from a certain point on is said to be eventually periodic. More precisely, a sequence $$(x_1,x_2,\ldots )$$ is eventually periodic if for some $$k\in \mathbb {N}$$ the sub-sequence $$(x_k,x_{k+1},\ldots )$$ is periodic. If the periodic pattern consists of a single term, then the sequence is said to be eventually constant.

The sequence of digits of a rational number is eventually periodic.

The idea of continuity does not apply to sequences because the concept of neighbourhood of an integer is pointless. Except at infinity, that is. The neighbourhoods of infinity are the infinite sets of the form$$\{k\in \mathbb {N}\,:\, k>K\}$$, for some integer $$K$$, and we use the expression for all sufficiently large $$k$$, to mean for all $$k>K$$, for some $$K$$.

We can now test continuity at infinity. Adapting Definition (5.10) to the present situation, we say that the sequence $$(a_k)$$ converges to the limit $$c$$ if, given any neighbourhood $$J$$ of $$c$$, all terms of the sequence eventually belong to $$J$$. In symbols:

$$ \forall J \in {\fancyscript{N}}_c,\,\,\exists K\in \mathbb {N},\,\,\forall k>K,\,\,a_k\in J. $$

This expression is written concisely as

$$ \lim _{k\rightarrow \infty }a_k=c. $$

By stipulating that $$a_\infty =c$$, we see that a real sequence converges if it’s continuous at infinity. In the special case $$c=\infty $$ we say that the sequence diverges.

Exercise 5.1

Write each symbolic sentence in two ways:

(i)

(ii)

Exercise 5.2

For each symbolic sentence of the previous exercise, give examples of functions for which the sentence is true and functions for which the sentence is false.

Exercise 5.3

Write each symbolic sentence without symbols, apart from $$f$$.

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Exercise 5.4

Turn words into symbols.

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Exercise 5.5

Consider the following implications, where $$f$$ is a real function.

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Of each implication:

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Exercise 5.6

As in Exercise 4.3.

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Exercise 5.7

Consider the Definition (5.5) of a periodic function. What happens if we place the quantifiers in reverse order?

$$\begin{aligned}&\forall x \in \mathbb {R},\,\,\exists T\in \mathbb {R}^*,\,\,f(x+T)=f(x)\end{aligned}$$

(5.14)

$$\begin{aligned}&\forall x \in \mathbb {R},\,\,\exists T\in \mathbb {R}^+,\,\,f(x+T)=f(x). \end{aligned}$$

(5.15)

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Exercise 5.8

Describe the behaviour of the following functions. [ $$/\!\!\varepsilon ,30$$ ]

Exercise 5.9

By experimenting on a computer, if necessary, define real functions whose behaviour is qualitatively similar to those of the previous problem.

Exercise 5.10

Give an example of a real function with the stated properties.

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Exercise 5.11

Express each statement with symbols.

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Exercise 5.12

Find a representation for the abstract set $${\fancyscript{N}}_x$$ of all neighbourhoods of a point $$x\in \mathbb {R}$$ (see Sect. 2.3.3).

Footnotes

1

This expression indicates that an inessential restriction or simplification is being introduced. See Sect. 7.8 for another example.

2

The term ’interval’ is intended in the proper sense—infinite intervals are excluded, cf. (2.13).