Sequence
A sequence is an ordered list of terms. Sequences may be either finite or infinite.
Definition
A sequence of real numbers is simply a function . For instance, the function
defined on
corresponds to the sequence
.
Convergence
Intuitively, a sequence converges if its terms approach a particular number.
Formally, a sequence of reals converges to
if and only if for all positive reals
, there exists a positive integer
such that for all integers
, we have
.
If converges to
,
is called the limit of
and is written
. The statement that
converges to
can be written as
.
Monotone Sequences
Many significant sequences are increasing or decreasing. This motivates the following definitions:
A sequence of reals is said to be
- increasing if
for all
and strictly increasing if
for all
,
- decreasing if
for all
and strictly decreasing if
for all
,
- monotone if it is either decreasing or increasing.
Resources
See Also
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