Question

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

\({a_n} = \frac{{{{\tan }^{ - 1}}n}}{n}\)

Short answer

Converges

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\({a_n} = \frac{{{{\tan }^{ - 1}}n}}{n}\)

Using\(\mathop {\lim }\limits_{n \to \infty } {\tan ^{ - 1}}n = \frac{\pi }{2}\)

So, we have:

\(\begin{aligned}\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{{{\tan }^{ - 1}}n}}{n}} \right) &= \frac{{\frac{\pi }{2}}}{\infty }\\ &= 0\end{aligned}\)

So the sequence converges to zero as\(n \to \infty \).