**Given:**

The sequence is
an=3+5n21+n
.

**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→∞3+5n21+n
.

Divide the numerator and the denominator by the highest power.

limn→∞3+5n21+n=limn→∞3+5n2n1+nn=limn→∞3n+5n2n1n+nn

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

limn→∞3+5n21+n=limn→∞3n+5n1n+1=limn→∞(3n)+limn→∞(5n)limn→∞(1n)+limn→∞(1)=3⋅limn→∞(1n)+5⋅limn→∞(n)limn→∞(1n)+1

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

limn→∞3+5n2n+n2=3⋅(1∞)+5⋅∞(1∞)+1=0+∞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.