Unbounded sequence

Definition

An unbounded sequence is a sequence that does not converge to any finite limit, meaning its terms grow without bound as they approach infinity. In mathematical terms, for any real number $M$, there exists an index $N$ such that for all $n \geq N$, the absolute value of the sequence's terms is greater than $M$.

5 Must Know Facts For Your Next Test

1. An unbounded sequence can tend to positive or negative infinity.
2. If a sequence is unbounded, it cannot be convergent.
3. The formal definition involves proving that for any given real number $M$, there exists an index beyond which all terms of the sequence exceed $M$ in absolute value.
4. Common examples of unbounded sequences include arithmetic and geometric sequences with common ratios greater than one.
5. Unboundedness implies that the terms of the sequence will eventually lie outside any finite interval.

Review Questions

• What distinguishes an unbounded sequence from a bounded sequence?
• How would you formally prove that a given sequence is unbounded?
• Can an unbounded sequence ever be convergent? Explain why or why not.

Related terms

Bounded Sequence: A sequence whose terms lie within a fixed interval. There exist real numbers $m$ and $M$ such that every term of the sequence is between $m$ and $M$.

Convergent Sequence: A sequence whose terms approach a specific value (the limit) as the number of terms goes to infinity. Formally, for every $\epsilon > 0$, there exists an index $N$ such that for all $n \geq N$, the absolute difference between the term and the limit is less than $\epsilon$.

Divergent Sequence: A sequence that does not converge to any finite limit. This includes sequences that are unbounded or oscillate without settling to a single value.