Final answer:
The sequence an = n³ / (2n + 7) diverges as n approaches infinity because the highest power in the numerator is larger than that in the denominator, indicating that an grows without bound.
Step-by-step explanation:
To determine whether the sequence an = n³ / (2n + 7) converges or diverges, we examine the behavior of the sequence as n approaches infinity. In this case, we can compare the degree of the polynomial in the numerator and the denominator. Since the highest power in the numerator is n³ and in the denominator is 2n, we can see that the numerator grows much faster than the denominator as n becomes very large. Therefore, the sequence diverges to infinity, and there is no finite limit.
Another way to formally prove this is by using L'Hôpital's Rule, which states that if the limit of f(n)/g(n) as n approaches infinity is of the form 0/0 or ∞/∞, then the limit of f(n)/g(n) is the same as the limit of f'(n)/g'(n).
In our case, even without calculating derivatives, it is clear that the sequence diverges because the highest powers in the numerator and denominator are not the same. Thus, the limit of an as n approaches infinity does not exist.