Final answer:
The series does not converge absolutely, but we need to perform another test to determine whether it converges conditionally or diverges.
Step-by-step explanation:
The series ∑ₙ=1[∞](-1)ⁿ+15+n⁴/n⁵ represents a series with alternating signs and increasing powers of n as terms. To determine whether it converges absolutely, converges conditionally, or diverges, we can use the Alternating Series Test. For an alternating series to converge, the absolute values of the terms must decrease and approach zero as n approaches infinity. In this case, the term (-1)ⁿ+15+n⁴/n⁵ does approach zero as n approaches infinity. However, to determine if the terms are decreasing, we need to analyze the ratio of consecutive terms.
Let's consider the ratio of the (n+1)th term to the nth term:
|((-1)ⁿ⁺¹ + 15 + (n+1)⁴⁺¹/ⁿ⁺¹⁵) / ((-1)ⁿ + 15 + n⁴/ⁿ⁺⁵)|
Simplifying the expression and taking the limit as n approaches infinity, we find that the ratio converges to 1. Since the ratio does not approach zero, the terms of the series do not decrease, and thus the series does not converge absolutely. However, it is still possible for the series to converge conditionally. We need to check whether the series formed by the absolute values of the terms converges or diverges by using a different convergence test.