An unbounded sequence is a sequence which is not bounded. For example the sequence {n} of natural numbers is unbounded. In this article, we study unbounded sequences with its definition and examples.
Unbounded Sequence Definition
A sequence {an} is said to be unbounded if there is no real number M such that
|an| ≤ M.
For example, the sequence {n} is unbounded.
Unbounded Sequence Example
Question 1: Prove that the sequence {2n} is unbounded.
Answer:
For a contradiction, assume that the sequence {(-1)n n} is bounded. So there is a real number M such that
|an| < M for all natural number n. Here an denotes the nth term of the sequence, that is, an = (-1)n n.
⇒ |(-1)n n| ≤ M ∀ n ∈ ℕ
⇒ n ≤ M ∀ n ∈ ℕ.
But for the natural number [M]+2, this inequality fails. Here, [x ] denotes the greater integer function of x.
Hence, our assumption was wrong. In other words, we conclude that the sequence {(-1)n n} is unbounded.
Question 1: Prove that the sequence {(-1)n n} is unbounded.
Answer:
The n-th term of the sequence is an = 2n.
Let us assume that the sequence {2n} is bounded. Thus, by definition, there is a real number M such that
|an| < M for all natural number n
⇒ |2n| ≤ M ∀ n ∈ ℕ
⇒ 2n ≤ M ∀ n ∈ ℕ …(i)
Consider the natural number n=[M], where [x ] denotes the greater integer function of x. For this natural number, this above inequality (i) fails, that is 2[M] ≤ M is not true.
Hence, our assumption was wrong. In other words, we conclude that the sequence {(-1)n n} is unbounded.
Related Articles:
Bounded sequence definition examples
A Convergent Sequence is Bounded: Proof, Converse
FAQs
Q1: What is an unbounded sequence? Give an example.
Answer: If a sequence {an} is not both bounded below and above, then it is called an unbounded sequence. That is, there are no real numbers k and K such that k ≤ an ≤ K ∀ n ∈ ℕ.
For example, the sequence {2n} is not bounded.