Question

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

\({\bf{\{ 0,1,0,0,1,0,0,0,1, \ldots \ldots \ldots \ldots \ldots \} }}\)

Short answer

Diverges

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\(\{ 0,1,0,0,1,0,0,0,1, \ldots \ldots \ldots \ldots \ldots \} \)

Since the sequence takes only two values 0 and 1.

Therefore the sequence\({a_n}\)does not approach any single number as the number of terms increases. So it is divergent.