Definition:Divergent Sequence

From ProofWiki
Jump to navigation Jump to search

This page is about divergent sequences. For other uses, see Divergent (Analysis).

Definition

A sequence which is not convergent is divergent.


This article is complete as far as it goes, but it could do with expansion.
In particular: Explain how this specifically applies in the different contexts: Topological space, metric space, etc.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Divergent Real Sequence

A real sequence which is not convergent is divergent.


Examples

Example: $\paren {\dfrac 2 3 + \dfrac {3 i} 4}^n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\dfrac 2 3 + \dfrac {3 i} 4}^n$

Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.


Example: $\paren {-1}^n + \dfrac i n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n \paren {-1}^n + \dfrac i n$

Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.


Example: $i^n$

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = i^n$

Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.


Also see

  • Results about divergent sequences can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Divergent_Sequence&oldid=713457"