Augustin-Louis Cauchy

cauchy:front

(Belliard, 1901)

1 - Introduction

A sequence is an ordered list of numbers

where each term $a_1$, $a_2$, $a_3$, ..., $a_n$, ... represents a number.

The position of a term in a sequence is important. Therefore, we assign a subscript $n$ to each term $a_n$ to mean $a_n$ occupies the $n$th position in the sequence. In this manner, term $a_1$ falls into position $1$, term $a_2$ falls into position $2$, term $a_n$ falls into position $n$, and so forth.

Figure 1. A term's subscript denotes the term's position in the sequence.

An infinite sequence contains an infinite number of terms.

For convenience, you can describe the assignment of each subscript $n$ to a term $a_n$ as a function that maps $(\rightarrow)$ subscript $n$ to exactly one term $a_n$ in the sequence.

Figure 2. A sequence is a mapping from subscripts to terms.

Suppose your sequence is the ordered list of even numbers

then your function $f$ is an even function

for all positive integers $n.$

Suppose instead your sequence is the ordered list of odd numbers

then your function $f$ is an odd function

for all positive integers $n.$

You can describe an infinite sequence in several ways. First, you can enumerate the infinite sequence as a list that includes an ellipsis, e.g.,

Second, you can describe the sequence as a rule for the $n$th term, e.g.,

And third, you can use set notation with a rule for the $n$th term, e.g.,

1.1 - Infinite sequence

An infinite sequence (Bonar & Khoury, 2006, p. 1) is a function from the nonnegative integers to the reals,

We think of a sequence as an ordered infinite list,

and write

or simply

For example, the ordered list of natural numbers is an infinite sequence,

where the list begins with the number one $(1)$ and continues indefinitely $(...)$ as a succession of terms, one term after the other.

The ordered list of even numbers beginning with the number two $(2)$ is also an infinite sequence,

And finally, the ordered list of odd numbers beginning with the number one $(1)$ is an infinite sequence,

References

Not to use ... for now

A sequence is an ordered list of terms

where each term denotes a number and each term's index denotes that term's position in the list.

We use the $n$th term enclosed in braces to denote the sequence in general,

where an indexed term $a_n$ is a number and the index $n$ is that term's position in the list.

An infinite sequence is an ordered infinite list of terms

written simply as

where an indexed term $a_n$ is a number and the index $n$ is that term's position in the list.

A finite sequence is a sequence that contains a fixed number of terms.

An infinite sequence is a sequence that contains an infinite number of terms.

A sequence is an enumerated collection of objects — numbers in our case — in which repetition is allowed and order matters. For example, the ordered collection of the first 5 natural numbers is a most basic example of a sequence,

where each number in the sequence is separated from the next using a comma. The ellipse (...) denotes the sequence continues indefinitely following along the same pattern.

The ordered collection of even numbers beginning with the number 2 is another example of a sequence,

Similarly, the ordered collection of odd numbers beginning with the number 1 is yet another example of a sequence,