**Given:**

The sequence is
an=3+5n2n+n2
.

**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+5n2n+n2
.

Divide the numerator and the denominator by the highest power.

limn→∞3+5n2n+n2=limn→∞3+5n2n2n+n2n2=limn→∞3n2+5n2n2nn2+n2n2

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

limn→∞3+5n2n+n2=limn→∞3n2+51n+1=limn→∞(3n2)+limn→∞(5)limn→∞(1n)+limn→∞(1)=3⋅limn→∞(1n2)+5limn→∞(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+50+1=51=5

Since
limn→∞an
is exist and by using the definition of the sequence, it can be concluded that the sequence is converges to the limit 5.

Therefore, the sequence is converges to the limit 5.