Definition:Divergent Sequence
(Redirected from Definition:Divergent Complex Sequence)
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. |
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
- Because a Convergent Sequence is Bounded, it follows directly that an unbounded sequence is divergent.
- However, Divergent Sequence may be Bounded shows that the converse does not necessarily hold.
- Results about divergent sequences can be found here.
Sources
- 1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): $\S 12$-$2$: Convergent and divergent sequences: Definition $12$-$1$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.26$: Divergent sequences
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): diverge: 1a.
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2$: Infinite Series of Constants
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divergence: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divergent sequence
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divergence: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divergent sequence
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): diverge