Editing Sequence (section)

===Indexing===
Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of [[pi|{{pi}}]]. One such notation is to write down a general formula for computing the ''n''th term as a function of ''n'', enclose it in parentheses, and include a subscript indicating the set of values that ''n'' can take. For example, in this notation the sequence of even numbers could be written as <math display=inline>(2n)_{n\in\mathbb N}</math>. The sequence of squares could be written as <math display=inline>(n^2)_{n\in\mathbb N}</math>. The variable ''n'' is called an [[Indexed family|index]], and the set of values that it can take is called the [[index set]].

It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like <math display=inline>(a_n)_{n\in\mathbb N}</math>, which denotes a sequence whose ''n''th element is given by the variable <math>a_n</math>. For example:
:<math>\begin{align}
a_1 &= 1\text{st element of }(a_n)_{n\in\mathbb N} \\
a_2 &= 2\text{nd element } \\
a_3 &= 3\text{rd element } \\
&\;\;\vdots \\
a_{n-1} &= (n-1)\text{th element} \\
a_n &= n\text{th element} \\
a_{n+1} &= (n+1)\text{th element} \\
&\;\; \vdots
\end{align}</math>
One can consider multiple sequences at the same time by using different variables; e.g. <math display=inline>(b_n)_{n\in\mathbb N}</math> could be a different sequence than <math display=inline>(a_n)_{n\in\mathbb N}</math>. One can even consider a sequence of sequences: <math display=inline>((a_{m, n})_{n\in\mathbb N})_{m\in\mathbb N}</math> denotes a sequence whose ''m''th term is the sequence <math display=inline>(a_{m, n})_{n\in\mathbb N}</math>.

An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation <math display=inline>(k^2){\vphantom)}_{k = 1}^{10}</math> denotes the ten-term sequence of squares <math>(1, 4, 9, \ldots, 100)</math>. The limits <math>\infty</math> and <math>-\infty</math> are allowed, but they do not represent valid values for the index, only the [[supremum]] or [[infimum]] of such values, respectively. For example, the sequence <math display=inline>{(a_n)}_{n = 1}^\infty</math> is the same as the sequence <math display=inline>(a_n)_{n\in\mathbb N}</math>, and does not contain an additional term "at infinity". The sequence <math display=inline>{(a_n)}_{n = -\infty}^\infty</math> is a '''bi-infinite sequence''', and can also be written as <math display=inline>(\ldots, a_{-1}, a_0, a_1, a_2, \ldots)</math>.

In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes <math display=inline>(a_k)</math> for an arbitrary sequence. Often, the index ''k'' is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in
:<math>{(a_k)}_{k=0}^\infty = ( a_0, a_1, a_2, \ldots ).</math>
In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of [[odd number]]s could be denoted in any of the following ways.

* <math>(1, 9, 25, \ldots)</math>
* <math>(a_1, a_3, a_5, \ldots), \qquad a_k = k^2</math>
* <math>{(a_{2k-1})}_{k=1}^\infty, \qquad a_k = k^2</math>
* <math>{(a_{k})}_{k=1}^\infty, \qquad a_k = (2k-1)^2</math>
* <math>\bigl((2k-1)^2\bigr)_{k=1}^\infty</math>

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the [[natural numbers]]. In the second and third bullets, there is a well-defined sequence <math display=inline>{(a_{k})}_{k=1}^\infty</math>, but it is not the same as the sequence denoted by the expression.