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

The sequence is converges to the limit π2 .

Given:

The sequence is an=arctan(lnn) .

Definition used:

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

Result used: Limit chain rule

If limu→bf(u)=L and limx→ag(x)=b with f(x) is continuous at x=b , then the value of limx→af(g(x)) is L.

Calculation:

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

Consider the functions g(n)=lnn and f(u)=arctan(u) .

The value of limn→∞g(n) is,

limn→∞g(n)=limn→∞(lnu)=ln∞=∞

The value of limu→∞f(u) is,

limu→∞f(u)=limu→∞(arctan(u))=arctan(∞)=π2

Apply the limit chain rule and obtain the limit.

Since g(n)=lnn and f(u)=arctan(u) , limn→∞f(lnn)=π2 and limn→∞(arctan(lnn))=π2 .

Therefore, limn→∞f(g(n))=π2 .

Since limn→∞an is exists and by using the definition, it can be concluded that the sequence is converges to the limit π2 .

Therefore, the sequence converges to the limit π2 .