Final answer:
The series ∑(4+4n)/(4+n)² is compared to the divergent series 1/n using the Limit Comparison Test. After simplification, the limit of the nth term ratio is found to be finite and non-zero, therefore the original series is divergent.
Step-by-step explanation:
The student asks whether the series ∑aₙ = ∑ (from n=1 to ∞) (4+4n) / (4+n)² is convergent or divergent using the Limit Comparison Test. To use this test, we compare the given series to a simpler series which we know converges or diverges. A natural choice for comparison is the series 1/n, which diverges.
To apply the Limit Comparison Test, we need to calculate the limit of aₙ/bₙ as n approaches infinity, where aₙ is the nth term of our given series, and bₙ is the nth term of the comparison series, which in this case is 1/n. So the limit is:
limit (n → ∞) of [(4+4n)/(4+n)²] / (1/n)
After simplifying the expression and using algebra, we find the limit as n approaches infinity is 4. Since the limit is finite and non-zero, and the comparison series 1/n diverges, our initial series also diverges by the Limit Comparison Test.