**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
.