Convergent Sequence: Definition, Examples

A convergent sequence is a sequence having finite limit. For example, {1/n} is a convergent sequence with limit 0. Here, we study the definition and examples of convergent sequence.

Convergent Sequence Definition

A sequence {x_n} is said be convergent if it has a finite limit L. That is, for every ε>0, there is a natural number N such that

|xn – L| < ε for all n ≥ N.

We say the sequence {x_n} converges to L. Symbolically, we write lim_n→∞ x_n = L.

For example, the sequence {1/n} is convergent and it converges to 0.

Remark: If a sequence is not convergent then it is either divergent or oscillatory. For example, the sequence {n} is divergent.

Convergent Sequence Examples

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

Answer:

Here, the nth term is $x_n=\dfrac{1}{n^2}$.

Let ε > 0 be arbitrary.

Now, |x_n – 0|

= $|\dfrac{1}{n^2} -0|$

= $\dfrac{1}{n^2}$ < ε if $n > \dfrac{1}{\sqrt{\epsilon}}$

Chose N = $[\dfrac{1}{\sqrt{\epsilon}}]$+1 where [x] denotes the greatest integer function of x.

Therefore, we have shown that

$|\dfrac{1}{n^2} -0|$ < ε if n≥N.

Thus, by definition, the sequence {1/n²} converges, and it converges to 0.

Note: In a similar way, one can show that the sequence {1/n^p} (where p>0) converges to 0, hence convergent.

More Problems:

• Show $\left\{ \dfrac{1}{n+1}\right\}$ is convergent.
• Show {xⁿ} converges to 0 provided that |x| < 1.
• The sequence {n} is not convergent.

Related Articles:

A Convergent Sequence is Bounded: Proof, Converse

Bounded sequence definition examples

Unbounded sequence definition examples

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