Divergent Sequence: Definition, Examples

A divergent sequence is a sequence having limit either infinite (that is, does not converge to a specific finite limit). For example, the sequence {n} has infinite limit, hence a divergent sequence. Here we learn the definition and examples of divergent sequence.

Divergent Sequence Definition

A sequence {x_n} is said be divergent if its limit is infinite. That is, a sequence can diverge to either +∞ or -∞. For example, the sequence {n} diverges to +∞. So {n} is a divergent sequence.

Mathematically, a sequence {x_n} diverges to +∞ if for any positive integer M we can find a N ∈ ℕ such that

xn > M for all n ≥ N.

In this case, we write lim_n→∞ x_n = +∞.

Similarly, one can define a sequence that diverges to -∞. For example, the sequence {-n} diverges to -∞.

Divergent Sequence Examples

Question 1: Show that the sequence {n²} is divergent.

Answer:

We have x_n = n² (the nth term).

Note that n² > k if n > √k.

Thus, for a positive integer M, we can choose N = [√M]+1 ∈ ℕ such that

n² > M for all n > N.

Hence, we obtain that

xn > M for all n > N.

Thus by the above definition, the sequence {n²} diverges to +∞. Therefore, {n²} is a divergent sequence.

Note: In a similar way, one can show that the sequence {n^p} (where p>1) diverges to +∞, hence it is a divergent sequence.

Question 2: Show that the sequence {-n} is divergent.

Answer:

In a similar way as Q1, we can show that the sequence {-n} diverges to -∞, hence it is a divergent sequence.

Related Articles:

A Convergent Sequence is Bounded: Proof, Converse

Convergent Sequence: Definition and Examples

Bounded sequence definition examples

Unbounded sequence definition examples

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