Difference between revisions of "Sequence"
(Added section on monotone sequences; we don't need the wiki pages for these anymore.) |
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==Monotone Sequences== | ==Monotone Sequences== | ||
− | Many significant sequences | + | Many significant sequences have their terms continually increasing, such as <math>(n^2)</math>, or continually decreasing, such as <math>(1/n)</math>. This motivates the following definitions: |
A sequence <math>(p_n)</math> of reals is said to be | A sequence <math>(p_n)</math> of reals is said to be |
Revision as of 15:07, 17 October 2012
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 have their terms continually increasing, such as , or continually decreasing, such as
. 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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