Final answer:
To determine the convergence or divergence of the given sequence, we would apply the Alternating Series Test or the Comparison Test. The series might converge if it is an alternating series and if the polynomial's terms in the denominator approach zero as n goes to infinity. However, the correct application depends on the precise definition of the series, which seems to have a potential typo in the question.
Step-by-step explanation:
The question at hand involves determining the convergence or divergence of a given sequence a₍ₙ, which is defined as a₍ₙ=(-1)ⁿ¹/n² + 7 n + 12. To ascertain whether this sequence converges or diverges, we need to analyze its behavior as n approaches infinity.
The sequence appears to be the sum of an alternating series and a polynomial term. Typically, for sequences of this nature, we would use either the Alternating Series Test or the Comparison Test to assess convergence.
However, based on the information provided, it is difficult to provide a detailed method without more context or a clearer definition of the sequence. The notation seems to suggest that the sequence could be an alternating series, but the '+1' appended to the power of -1 is unusual and could be a typo.
In this hypothetical case, because the denominator is a second-degree polynomial, as n goes to infinity, the terms of the series would indeed approach 0, potentially satisfying the second condition of the Alternating Series Test.
To satisfy the first condition, one would have to show that |a₍ₙ+1| < |a₍ₙ| for all n, which in this case is likely due to the behavior of the polynomial as n increases. If both conditions are met, the series would converge; otherwise, it would diverge.