Whether the sequence is converges or diverges and obtain the limit if the sequence is converges.

The sequence is diverges.

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.