Number Sequence

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In physics and other sciences there are many quantities: time, length, volume etc. Some of them are variable (take different values) and some are constant (take constant values).

However, in mathematics we abstract from physical sense of quantity, we are interested only in numbers that this quantity can take. Physical sense becomes important in applications of mathematics. Therefore, for us variable quantity (or simple variable) is number variable. It is denotes by some symbol (for example, $x$) to which we attach numeric value.

Variable is specified if there is set ${X}={\left\{{x}\right\}}$ of values that it can take. It is convenient to regard constant quantity as special case of variable quantity; in this case set ${X}={\left\{{x}\right\}}$ consists of one element.

It is not enough to know from which set ${X}$ variable takes values, we also need to know what exactly values (among which can be repeating) and in what order does it take.

Consider list of natural numbers ${1},{2},{3},\ldots,{n},\ldots,{n}',\ldots,$ in which numbers are placed in increasing order, so that bigger number ${n}'$ is after smaller number ${n}$. If we now substitute by some law each natural number in sequence with some real number ${x}_{{n}}$, then we will obtain numerical sequence ${x}_{{1}},{x}_{{2}},{x}_{{3}},\ldots,{x}_{{n}},\ldots,{x}_{{{n}'}},\ldots,$ elements of which ${x}_{{n}}$ are numbered with natural numbers and placed in increasing order. When ${n}'>{n}$ ${x}_{{{n}'}}$ is after ${x}_{{n}}$ regardless of whether number ${x}_{{{n}'}}$ is greater, lesser or even equal ${x}_{{n}}$.

Definition. Sequence is an ordered list of objects in which if ${n}'>{n}$ then ${x}_{{{n}'}}$ is after ${x}_{{n}}$ regardless of what values ${x}_{{{n}'}}$ and ${x}_{{n}}$ actually take.

In other words we set in correspondence to every natural number some element, not always number. For example, ${a},{b},{c},{d},{e},{f},\ldots,{x},{y},{z}$ is a finite sequence of letters of English alphabet. Here ${a}$ corresponds to number 1 (${a}$ is on first place): ${x}_{{1}}={a}$, ${b}$ corresponds to number 2 (${b}$ is on second place): ${x}_{{2}}={b}$ etc.

However, in calculus we handle only numerical sequence, i.e. sequences that contain only real numbers.

Example of numerical sequence is ${1},\frac{{1}}{{2}},\frac{{1}}{{3}},\frac{{1}}{{4}},\ldots$. Here, first element is ${x}_{{1}}={1}$, second element is ${x}_{{2}}=\frac{{1}}{{2}}$ etc.

Sequences can be finite and infinite. Example with alphabet is example of finite sequence, because we can count number of elements that it contains. Infinite sequence contains infinite number of members, for example, ${1},{2},{3},{4},{5},{6},{7},{8},{9},{10},\ldots$.

Sequences can be denoted in different ways: ${x}_{{n}},{\left\{{x}_{{n}}\right\}},{{\left\{{x}_{{n}}\right\}}_{{{n}={1}}}^{{\infty}}}$. Also we can just list elments inside of curly braces like ${\left\{{1},\frac{{1}}{{2}},\frac{{1}}{{3}},\frac{{1}}{{4}},\frac{{1}}{{5}},\ldots\right\}}$.

Now, let's see more common examples of sequences.

Example 1. Arithmetic progression is an example of sequence: ${x}_{{1}}={a},{x}_{{2}}={a}+{d},{x}_{{3}}={a}+{2}{d},\ldots,{x}_{{n}}={a}+{\left({n}-{1}\right)}{d},\ldots$.

Example 2. Geometric progression is an example of sequence: ${x}_{{1}}={a},{x}_{{2}}={a}{q},{x}_{{3}}={a}{{q}}^{{2}},\ldots,{x}_{{n}}={a}{{q}}^{{{n}-{1}}},\ldots$.

Example 3. Sequence of decimal approximations of $\sqrt{{{2}}}$ with increasing precision: ${x}_{{1}}={1.4},{x}_{{2}}={1.41},{x}_{{3}}={1.414},\ldots$.

Variable ${x}$ that takes value from sequence is often denoted by ${x}_{{n}}$ (just as common variable member of this sequence).

Sometimes variable ${x}$ is specified by specifying expression for ${x}_{{n}}$; in case of arithmetic progression ${x}_{{n}}={a}+{\left({n}-{1}\right)}{d}$, and in case of geometric progression ${x}_{{n}}={a}{{q}}^{{{n}-{1}}}$.

In general n-th member of sequence can be defined by some formula ${x}_{{n}}={f{{\left({n}\right)}}}$. Using this formula we can calculate any member of sequence using its number, without calculating previous values. This is called analytical definition of sequence. For example, if sequence is defined by formula ${x}_{{n}}={1}+\frac{{{{\left(-{1}\right)}}^{{n}}}}{{n}}$ then ${x}_{{3}}={1}+\frac{{{{\left(-{1}\right)}}^{{3}}}}{{3}}={1}-\frac{{1}}{{3}}=\frac{{2}}{{3}}$.

In other cases there can be no expression for ${x}_{{n}}$. Nevertheless, sequence is specified if we have a rule, by which we can calculate any member by its number. Therefore, knowing rule for approximate calculation of roots, we can consider specified all sequence of decimal approximations of $\sqrt{{{2}}}$, even without knowing expression for ${x}_{{n}}$.

If sequence is specified, then we know not only values that it can take, but also order in which these values are taken.

It is worth noting, that values in sequence are not required to be different.

Example 4. Consider three sequences ${x}_{{n}}={1}$, ${x}_{{n}}={{\left(-{1}\right)}}^{{{n}+{1}}}$, ${x}_{{n}}=\frac{{{1}+{{\left(-{1}\right)}}^{{n}}}}{{n}}$.

If we write them as list, then we will obtain following:

{1, 1, 1, 1, 1, 1, 1, 1, ...},

{1, -1, 1, -1, 1, -1, 1, -1, ...},

${\left\{{0},{1},{0},\frac{{1}}{{2}},{0},\frac{{1}}{{3}},\ldots\right\}}$.

In first case we have just constant value, set of values that it can take consists of one element, namely 1.

In sequence can take only two values: 1 and -1.

In third case set of taken values is infinite. Howerver, we see that takes takes more than one time value 0.

So, sequence can be defined in three ways:

1. Analytical: we can find n-th member with the formula that depends on n, for example, ${x}_{{n}}=\frac{{2}}{{n}}$. In this case ${x}_{{1}}=\frac{{2}}{{1}}={2}$, ${x}_{{2}}=\frac{{2}}{{2}}={1}$, ${x}_{{3}}=\frac{{2}}{{3}}$ etc. So we can rewrite sequence in the form of list: ${\left\{{2},{1},\frac{{2}}{{3}},\ldots\right\}}$.
2. Recursive: n-th member of sequence can be found based on values of previous members. For example, Fibonacci sequence is defined as ${x}_{{1}}={1}$, ${x}_{{2}}={1}$, ${x}_{{n}}={x}_{{{n}-{1}}}+{x}_{{{n}-{2}}},{\left({n}\ge{2}\right)}$. So, sequence is ${1},{1},{2},{3},{5},{8},{13},{21},{34},\ldots$. In this sequence each member, except first two, is sum of two previous members.
3. Verbal: we define sequence by describing it. For example, sequence where n-th member is number $\sqrt{{{2}}}$ correct to n decimal places: ${1.4},{1.41},{1.414},\ldots$.