Definition:Divergent Sequence

From ProofWiki
Jump to navigation Jump to search

Definition

A 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 $\displaystyle \lim_{n \mathop \to \infty} z_n$ 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 $\displaystyle \lim_{n \mathop \to \infty} z_n$ does not exist.


Example: $i^n$

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

$z_n = i^n$

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


Also see


Sources