**Given:**

The sequence is
an=n2n3+4n
.

**Definition used:**

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

**Laws of limits for sequences used:**

If
f(x)
and
g(x)
are two functions, then
limx→a[f(x)g(x)]=limx→af(x)limx→ag(x)
and
limx→ag(x)≠0
.

**Calculation:**

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

Compute
limn→∞n2n3+4n
.

Divide the numerator and the denominator by the highest power.

limn→∞n2n3+4n=limn→∞n2n3n3+4nn3=limn→∞n⋅nnnn3+4nn3=limn→∞n⋅nnn3n3+4nn3=limn→∞n1+4n2

Apply the laws of limits for sequences and simplify the terms.

limn→∞n2n3+4n=limn→∞n121+4limn→∞(n2)=(limn→∞n)121+4(limn→∞n)2=∞1+0=∞

Since
limn→∞an
does not exist and by using the definition of the sequence, it can be concluded that the sequence is diverges.

Therefore, the sequence is diverges.