A convergence criterion is a criterion used to verify the convergence of a sequence.
Table of contents
The following is a commonly utilized convergence criterion.
Proposition
Let
![[eq1]](/images/convergence-criterion__1.png){kind=small}
be a sequence. Then,
![[eq2]](/images/convergence-criterion__2.png){kind=small}
where
![[eq3]](/images/convergence-criterion__3.png){kind=small}
is the distance between
and
.
In other words, a sequence is convergent to an element
if and only if the terms of the sequence become closer and closer to
when
is increased (closeness is measured by a distance function
).
Consider a sequence
of
-dimensional
vectors whose generic entry
is![[eq5]](/images/convergence-criterion__12.png){kind=small}
Because
and
converge to
as
becomes large, our intuition tells us that the sequence
should converge to the vector
defined as
follows:![[eq7]](/images/convergence-criterion__19.png){kind=small}
How
do we verify that this is indeed the case? First of all we need to define a
distance function
to measure the distance between
and a generic term of the sequence
.
If we use Euclidean
distance to measure the distance between
and
,
we
obtain![[eq8]](/images/convergence-criterion__25.png)
But
converges to
by increasing
.
Therefore, according to the convergence criterion above, the sequence
converges to
.
The lecture entitled Limit of a sequence provides more details about this convergence criterion.
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Please cite as:
Taboga, Marco (2021). "Convergence criterion", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/convergence-criterion.
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