# Infinite Sequences

Introduction to Sequences

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

It depends.

#### Explanation:

There are many types of sequences. Some of the interesting ones can be found at the online encyclopedia of integer sequences at https://oeis.org/

Let's look at some simple types:

Arithmetic Sequences

${a}_{n} = {a}_{0} + \mathrm{dn}$

e.g. $2 , 4 , 6 , 8 , \ldots$

There is a common difference between each pair of terms.

If you find a common difference between each pair of terms, then you can determine ${a}_{0}$ and $d$, then use the general formula for arithmetic sequences.

Geometric Sequences

${a}_{n} = {a}_{0} \cdot {r}^{n}$

e.g. $2 , 4 , 8 , 16 , \ldots$

There is a common ratio between each pair of terms.

If you find a common ratio between pairs of terms, then you have a geometric sequence and you should be able to determine ${a}_{0}$ and $r$ so that you can use the general formula for terms of a geometric sequence.

Iterative Sequences

After the initial term or two, the following terms are defined in terms of the preceding ones.

e.g. Fibonacci

${a}_{0} = 0$
${a}_{1} = 1$
${a}_{n + 2} = {a}_{n} + {a}_{n + 1}$

For this sequence we find: ${a}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{- n}}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$

There are many ways to make these iterative rules, so there is no universal method to provide an expression for ${a}_{n}$

Polynomial Sequences

If the terms of a sequence are given by a polynomial, then given the first few terms of the sequence you can find the polynomial.

e.g.

$\textcolor{red}{1} , 2 , 4 , 7 , 11 , \ldots$

Form the sequence of differences of these values:

$\textcolor{red}{1} , 2 , 3 , 4 , \ldots$

Form the sequence of differences of these values:

$\textcolor{red}{1} , 1 , 1 , \ldots$

Once you reach a constant sequence like this, pick out the initial terms from each sequence. In this case $1$, $1$ and $1$.

These form the coefficients of a polynomial expression:

a_n = color(red)(1)/(0!) + (color(red)(1)*n)/(1!) + (color(red)(1)*n(n-1))/(2!)

$= {n}^{2} / 2 + \frac{n}{2} + 1$

• It depends on the type of sequence.

If the sequence is an arithmetic progression with first term ${a}_{1}$, then the terms will be of the form:

${a}_{n} = {a}_{1} + \left(n - 1\right) b$
for some constant b.

If the sequence is a geometric progression with first term ${a}_{1}$, then the terms will be of the form:

${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$
for some constant $r$.

There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , \ldots$

Each term is the sum of the two previous terms.

The ratio of successive pairs of terms tends towards the golden ratio $\phi = \frac{1}{2} + \frac{\sqrt{5}}{2} \cong 1.618034$

The terms of the Fibonacci sequence are expressible by the formula:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{-} n}{\sqrt{5}}$ (starting with ${F}_{0} = 0$, ${F}_{1} = 1$)

In general an infinite sequence is any mapping from $\mathbb{N} \to S$ for any set $S$. It can be defined in any way you like.

Finite sequences are the same, except that they are mappings from a finite subset of $\mathbb{N}$ consisting of those numbers less than some fixed limit, e.g. $\left\{n \in \mathbb{N} : n \le 10\right\}$

• A sequence is an infinite set of points which are ordered and a mapping can be associated with with that set and the set of natural numbers.

The general notion of a sequence is that it is an infinite set with every element associated with a natural number even though all infinite sets may not be sequences.

A sequence may be represented by, $\left\{{x}_{n}\right\}$ where ${x}_{n}$ is the $n$th element related to the a corresponding natural number.

Thus if ${x}_{n} = \frac{1}{n} ^ 2$, the sequence may be given as,

$\left\{1 , \frac{1}{4} , \frac{1}{9} , \frac{1}{16} , \ldots .\right\}$

For ${x}_{n} = {n}^{3}$ we shall have,

$\left\{1 , 8 , 27 , 64 , \ldots .\right\}$

Now since, every set is Isomorphic to itself, for ${x}_{n} = n$ we can have,

$\left\{1 , 2 , 3 , \ldots . .\right\}$

This is indeed the set of naturals.

However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. They don't fit right with the definition I gave.

Let's for example take a Fibonacci sequence.

$1 , 1 , 2 , 3 , 5 , 8 , 13 , \ldots .$

This sequence is made by adding the previous two numbers on the list to form the next one and so on.

There can be arithmetic sequences, like

$2 , 8 , 14 , 20 , \ldots .$ which has first term $2$ and common difference $6$.

I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs.

See explanation.

#### Explanation:

I will address the common mistakes academics such as mathematics educators make with infinite sequences.

a. The first mistake is that they actually believe there is such a thing as an infinite sequence . They believe in completed or finished infinity, which is of course nonsense because no supertask is possible. That's where the fallacious Eulerian definition of infinite sum comes from, that is, $S = L i m S$.

So, you'll hear them using the Eulerian definition on the one hand to describe an "infinite sum" and on the other hand, you'll hear them saying things like there is "no last term" in an infinite sequence.

This confuses them and they never understand or realise that Newton made a very bad choice of words when he used the phrase "Serierum Infinitarum".

b. "Infinite series" are without exception always finite, that is, you only need a few terms to determine many of their properties, including the most important bounds property with respect to convergent series. From the given terms you can determine any ${n}^{t h}$ term.

c. Academics do not understand the difference between "for each" and "for all".

d. Academics don't realise that limits do not care if all the terms are there, or even there at all. For example,

$0.9 + 0.09 + 0.009 + \ldots$ has a limit of $1$. This limit does not care about all the terms being there because no infinite sum is possible.

$0.9 < 1$
$0.99 < 1$
$0.999 < 1$

etc.

e. The infinity symbol is misunderstood as representing actual infinity, whereas it doesn't even represent potential infinity.
All the theory of limits has nothing to do with infinity. The epileptic eight $\setminus \infty$ causes more confusion than understanding. Academics need to realise this is not a well-formed concept.

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