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

$a_{n}=\frac{\tan ^{-1} n}{n}$

Whether the sequence is converges or diverges and obtain the limit if the sequence is converges.

The sequence converges to the limit 0.

Given:

The sequence is an=tan−1(n)n.

Definition used:

If an is a sequence and limn→∞an is exists then the sequence an is said to be converges otherwise it is diverges.

Calculation:

Obtain the limit of the sequence to investigate whether the sequence is converges or diverges.

Compute limn→∞an=limn→∞(tan−1(n)n).

Apply the limit property limx→a[f(x)g(x)]=limx→af(x)limx→ag(x) when limx→ag(x)≠0 and simplify the numerator and the denominator.

limn→∞(tan−1(n)n)=limn→∞(tan−1(n))limn→∞(n)=tan−1(∞)∞=π2⋅0=0

Since limn→∞an exists and by using the definition of the sequence, it can be concluded that the sequence is converges to the limit 0.

Therefore, the sequence converges to the limit 0.