Question

Determine whether the sequence converges or diverges. If it converges, find the limit.

\({a_n} = \cos \left( {\frac{2}{n}} \right)\)

Short answer

Sequence converges and value of the limit is 1.

Step-by-step solution

Definition

A sequence\(\left\{ {{a_n}} \right\}\)has the limit \(L\)and we write \(\mathop {\lim }\limits_{n \to \infty } {a_n} = L\;\;or\;\;{a_n} \to L\)as\(n \to \infty \)if we can make the terms\({a_n}\)as close to \(L\) as we like by taking\(n\)sufficiently large. If \(\mathop {\lim }\limits_{n \to \infty } {a_n}\)exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Evaluate limit

Consider the sequence \({a_n} = \cos \left( {\frac{2}{n}} \right)\)

Let \(f(x) = \cos \left( {\frac{2}{x}} \right)\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to \infty } \cos \left( {\frac{2}{x}} \right) &= \cos \left( {\mathop {\lim }\limits_{x \to \infty } \frac{2}{x}} \right)\\ &= \cos (0)\\ &= 1\end{aligned}\)

Therefore, the given sequence\({a_n} = \cos \left( {\frac{2}{n}} \right)\) converges and value of the limit is 1.