Sequence
Contents
Definition
A sequence in a set is a function from the set of natural numbers to .
The way such a sequence is described is simply by listing the images of 1,2,3,... in the right order. Explicitly, for a function , the sequence can be written as:
The values are called the terms of the sequence. Specifically, the value is called the term.
For instance, the sequence given by the function can be written as:
Note that it is not possible to unambiguously describe a sequence (which is infinite) by just listing the first few terms (of which there are only finitely many), but the general idea behind listing the first few terms and putting the ellipses ("...") is that people are expected to figure out the most natural choice of function that fits the first few terms.
Notation
Instead of using the typical function notation with the input to the function in parentheses, sequences are typically notated using a subscript notation. The sequence is named by a letter, and individual terms of the sequence are denoted by that letter with a subscript used for the position (index). For instance, for a sequence denoted with letter , the first term is denoted , the second term is denoted , and the term is denoted . The sequence itself is written with the shorthand .
Terminology
Term | Meaning | In subscript notation for a sequence | In function notation for sequence given by a function |
---|---|---|---|
term of a sequence | the appropriate function value | is the term | is the term |
index or position of a term | the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. | the index of is . | the index of is . |
successor or "next term" | the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the term is the term. Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition. |
The successor to is | The successor to is |
predecessor or "previous term" | the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the term is the term.< Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition. The first term doesn't have a predecessor. |
The predecessor to is if . | The predecessor to is . |
repetition-free sequence | a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. | If with , then . | If with , then . |
constant sequence | a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other. | For all , | For all , . |
eventually constant sequence | a sequence for which there exists a natural number such that the part of the sequence beyond that point is constant | For with , | For with , |
periodic sequence | a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number such that for all natural numbers , the term equals the term. The smallest such is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1. | ||
eventually periodic sequence | a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers and such that for all natural numbers , the term equals the term. | for all satisfying | for all satisfying |
range of a sequence | the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. | The set | The set |
There are various other notions associated with sequences specifically in the context of sequences that take values in the real numbers, i.e., sequences of real numbers.
Term | Meaning |
---|---|
increasing sequence (sometimes called strictly increasing sequence to distinguish it from non-decreasing sequence) | If , the term is less than the term. Note that it suffices to check that each term is less than the next term. |
non-decreasing sequence (sometimes called weakly increasing sequence or monotonically increasing sequence) | If , the term is less than or equal to the term. Note that it suffices to check that each term is less than or equal to the next term. |
decreasing sequence (sometimes called strictly decreasing sequence to distinguish it from non-increasing sequence) | If , the term is greater than the term. Note that it suffices to check that each term is less than the next term. |
non-increasing sequence (sometimes called weakly decreasing sequence or monotonically decreasing sequence) | If , the term is greater than or equal to the term. Note that it suffices to check that each term is less than or equal to the next term. |
monotone sequence | Either non-decreasing or non-increasing |
bounded sequence | The range of the sequence is a bounded subset of the reals (here, bounded means bounded both from above and from below). Note that any constant or periodic sequence is bounded. Also, any convergent sequence (defined later) is bounded. |
sequence bounded from above | The range of the sequence is bounded from above in the reals. Note that any bounded sequence is bounded from above. Also, any non-increasing sequence is bounded from above. |
sequence bounded from below | The range of the sequence is bounded from above in the reals. Note that any bounded sequence is bounded from below. Also, any non-decreasing sequence is bounded from below. |
convergent sequence | a sequence that has a limit. |
Note that the notions of bounded and convergent are special in that both notions refer to eventual behavior only. A sequence is bounded if and only if it is eventually bounded, i.e., the first few finitely many terms do not affect whether or not the sequence is bounded. Similarly, a sequence is convergent if and only if it is eventually convergent.
Sequences indexed from zero onward
For a number of applications, particularly power series, it is useful to consider sequences that are indexed starting from zero. Such sequences can be thought of as functions from the set . All the definitions and concepts developed for sequences can be considered for sequences indexed from zero onward.
Operations on sequences
Pointwise operations
We can do various pointwise operations on sequences just as we do other types of pointwise combination of functions, such as a pointwise sum, difference, product, or quotient (assuming the second sequence has no zero term). Explicitly, for sequences and :
- The sum is the sequence whose term is
- For a real number , is the sequence whose term is .
- The difference is the sequence whose term is
- The product is the sequence whose term is
- The quotient is the sequence whose term is . Note that this sequence makes sense only if none of the s are zero.
Shift operations
A shift operation on a sequence takes its terms and moves them all to the left or right.
- A left shift operation moves all the terms a certain amount to the left. The original first few terms disappear. Explicitly, left shifting a sequence by a natural number means the term of the new sequence is . The original first terms have disappeared in the process.
- A right shift operation moves all the terms a certain amount to the right. This introduces the problem that the first few terms of the new sequence are undefined, and need to be specified separately. Explicitly, left shifting a sequence by a natural number means the term of the new sequence is . The first terms of the new sequence need to be specified separately.