This web page lists various terms used in this Sequences and Series
course. It is very much under construction

.

- abelian group
- A group for which the binary operation is commutative.
- a priori
- A useful Latin phrase referring to
deductions

and meaningfrom the point of view of correct deductive logic

. - aka
- Abbreviation for
also known as

. - Archimedean ordered field
- See a separate web-page.
- Bernoulli's inequality
- The inequality $(1+x{)}^{n}\u2a7e1+nx$, for $n\in \mathbb{N}$ and $x>-1$ in the reals. (In fact it works for any ordered field.) The proof is by induction on $n$ using axioms for an ordered field.
- bijection
- A function $f:A\to B$ which is both an
injection and a surjection.
Bijections are also called
one-to-one correspondences

. - bound
- A single value greater than or equal to all elements of a set or all values of a function (upper bound) or less than or equal to all elements of a set or all values of a function (lower bound). A function or set which has an upper bound is bounded above. A function or set which has an lower bound is bounded below. One that is bounded above and below is bounded. Do not confuse this with a common non-mathematical use of the word limit.
- canonical
- Meaning
according to a set of rules, or canon

. This word is used in mathematics to mean that the definition of the object in question is forced upon us in some way: either because it is the simplest or most natural such definition that works or more usually because it is the only such definition. - complete
- (Refering to an Archimedean ordered field.) See a separate web-page.
- convergence
- This is the central notion of this course, discussed in a separate web page for sequences and a further web page for series.
- dummy variable
- A variable in an expression that is thought to range over all
integers (or all values fomr some other set). For example,
$r$ in $\sum _{r=1}^{\infty}\frac{1}{{2}^{r}}$
or $n$ in
the sequence $({a}_{n}{)}_{n=1,2,\dots}$

. (Note for pedants: the $n$ in the common abbreviationthe sequence $\left({a}_{n}\right)$

is a dummy variable ranging over all positive integers. The $n$ inthe value ${a}_{{n}^{2}}$

has a specific value and is not a dummy, and the $n$th term in the sequence has been taken and squared.) - embedding
- A map or function taking a structure $A$ (such as a group, ring, field, etc.) into
another similar structure $B$, so that the image of $A$ (considered as
a
substructure

of $B$) looks exactly the same as $A$. Such a function will always be injective and preserve any binary operations present. - field
- See a separate web-page.
- group
- A set with a binary operation that is associative and has identity and inverses.
- identification
- When two structures look identical (such as a structure and the image of it via an
embedding) it often makes sense to regard the two structures
as really being the same. We say that we
identify

them. Such identifications are not strictly logically correct, because the twocopies

of the same object really are different copies, but the simplification gained is always worthwhile. Examples include identifying the integers with the copy of the integers in an ordered field. - infimum
- If $A\subseteq \mathbb{R}$ is nonempty any bounded below, there
is a
*greatest*$y\in \mathbb{R}$ such that $y\u2a7da$ for all $a\in A$. This $y$ (whose existence follows from completeness) is called the infimum of $A$ and often written $\text{inf}\left(A\right)$. - injection
- A function $f:A\to B$ such that $f\left(x\right)\ne f\left(y\right)$ for
all $x\ne y$ from $A$. Injections are also called
one-to-one functions

. - isomorphism
- A map or function taking a structure $A$ (such as a group, ring, field, etc.) exactly onto
another similar structure $B$, so that both $A$ (considered as
a
substructure

of $B$) and $B$ look exactly the same. In other words, an isomorphism is an embedding that is surjective as well as injective. - limit
- This is the central notion of this course, discussed
in a separate web page and almost everywhere else.
You must get out of the habit of using this word in any other
sense. For example, the common use of the word "limit" to mean
"bound" (as in "there is a limit to what I can do",
or "the motor is limited to speeds below 50mph") is
*never*used in mathematics and you must never use it in this way. - metric space
- A nonempty set $X$ with a distance function $d:X\times X\to \mathbb{R}$ satisfying the axioms: (a) $\forall x,y\in Xd(x,y)=0\leftrightarrow x=y$; and (b) the triangle inequality. A metric space is a useful general axiomatic structure for analysis which allows the notions of convergence and continuity to be studied in a more general setting, especially for more advanced courses.
- ordered field
- See a separate web-page.
- sequence
- A list of infinitely many numbers taken in order, ${a}_{1},{a}_{2},{a}_{3},\dots $. Do not confuse this with the idea of a series.
- series
- A summation of a list of infinitely many numbers
$\sum _{n=1}^{\infty}{a}_{n}$.
*Do not confuse this with the idea of a sequence*, despite the fact that the common non-mathematical meaning of "sequence" and "series" is identical. There are separate definitions of convergence for sequences and series, and separate theories for these with some important differences that you need to be aware of. - surjection
- A function $f:A\to B$ such that for all $y\in B$
there is $x\in A$ with $y=f\left(x\right)$. Surjections are also called
onto functions

. - supremum
- If $A\subseteq \mathbb{R}$ is nonempty and bounded above,
an equivalent form of the completeness axiom for the reals
says that there is a
*least*$x\in \mathbb{R}$ such that $a\u2a7dx$ for all $a\in A$. (Such $x$ may or may not be an element of $A$.) For instance, if $A$ is the set of terms in a monotonic nondecreasing sequence then the limit of the sequence is such an $x$. This $x$ is called the supremum of $A$ and often written $\text{sup}\left(A\right)$. - vanishing sequence
- Another name for a
*null sequence*, i.e., one that converges to zero.