**Given:**

The sequence is
an=n4n3−2n
.

**Definition used:**

If
an
is a sequence and
limn→∞an
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→∞n4n3−2n
.

Divide the numerator and the denominator by the highest power.

limn→∞n4n3−2n=limn→∞n4n3n3−2nn3=limn→∞nn3n3−2nn3

Redefine the terms of the numerator and the denominator as follows:

limn→∞n4n3−2n=limn→∞n1−2n2=limn→∞(n)limn→∞(1−2n2)=limn→∞(n)limn→∞(1)−limn→∞(2n2)=limn→∞(n)1−2⋅limn→∞(1n2)

Apply infinity property
limn→∞(kna)=0
and simplify the numerator and the denominator.

limn→∞n4n3−2n=∞1−2⋅(1∞)=∞1−0=∞1=∞

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.