Definition:Term of Sequence

From ProofWiki
Jump to navigation Jump to search

Definition

The elements of a sequence are known as its terms.


Let $\sequence {x_n}$ be a sequence.

Then the $k$th term of $\sequence {x_n}$ is the ordered pair $\tuple {k, x_k}$.


Index

Let $\sequence {x_n}$ be a sequence.

Let $x_k$ be the $k$th term of $\sequence {x_n}$.

Then the integer $k$ is known as the index of $x_k$.


Also defined as

Some sources gloss over the fact that a sequence is a mapping and define the terms to be the elements of the range of the sequence:

We call $x_n$ the $n$th term of the sequence.
-- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach: $\S 4.2$

However, this simplistic treatment lacks the precision of the definition provided here, and $\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse it.


Also known as

A term of a sequence is referred to by some sources as an element of the sequence.

Some sources refer to the general term of a sequence when expressing it by means of a rule.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Term_of_Sequence&oldid=736390"