Definition:Oscillating Sequence

Definition

Let $S$ be one of the standard number fields $\Q$, $\R$ or $\C$.

Let $\sequence {x_n}$ be a sequence in $S$.

Let $\sequence {x_n}$ be divergent.

Suppose $\sequence {x_n}$ is not unbounded.

That is, let:

$\neg x_n \to \infty$ as $n \to \infty$

Then $\sequence {x_n}$ is an oscillating sequence.

Examples

Arbitrary Example

The sequence $\sequence {x_n}$ where $x_n = \paren {-1}^n$ is an example of an oscillating sequence.

Also see

• Results about oscillating sequences can be found here.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.26$: Divergent Sequences
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divergent sequence
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): oscillating sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divergent sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): oscillating sequence