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 meaning from 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+)
1+
, for
and
-1
in the reals. (In fact it works for any ordered field.)
The proof is by induction on using axioms
for an ordered field.
- bijection
- A function :
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,
in
=1
12
or in
the sequence (
)
=1,2,
.
(Note for pedants: the in the common abbreviation
the sequence (
)
is a dummy variable
ranging over all positive integers. The in
the value
2
has a specific value and is
not a dummy, and the th term in the sequence has been taken
and squared.)
- embedding
- A map or function taking a structure (such as a group, ring, field, etc.) into
another similar structure , so that the image of (considered as
a
substructure
of ) looks exactly the same as . 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 two copies
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
is nonempty any bounded below, there
is a greatest
such that
for all
. This
(whose existence follows from completeness)
is called the infimum of and often written
inf()
.
- injection
- A function :
such that ()
()
for
all
from . Injections are also called
one-to-one functions
.
- isomorphism
- A map or function taking a structure (such as a group, ring, field, etc.) exactly onto
another similar structure , so that both (considered as
a
substructure
of ) and 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 with a distance
function
satisfying the axioms: (a)
,
(,)=0
=
; 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,
1,2,3,
. Do not
confuse this with the idea of a series.
- series
- A summation of a list of infinitely many numbers
=1
. 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 :
such that for all
there is
with =()
. Surjections are also called
onto functions
.
- supremum
- If
is nonempty and bounded above,
an equivalent form of the completeness axiom for the reals
says that there is a least
such that
for all
. (Such may or
may not be an element of .) For instance, if
is the set of terms in a monotonic nondecreasing sequence
then the limit of the sequence is such an . This
is called the supremum of and often written
sup()
.
- vanishing sequence
- Another name for a null sequence, i.e., one that converges
to zero.