Final answer:
The alternating series presented in the question does not converge because the nth term does not approach zero, which violates the convergence criteria of the alternating series test.
Step-by-step explanation:
The question asks to verify if an alternating series converges and to find its sum using the alternating series test. The series in question is ∑[k=1 to ∞] (k(k+4) / 2) (-1)^(k+1). However, upon closer examination, the nth term of the series does not approach zero as n approaches infinity, which violates one of the criteria for an alternating series to converge. Therefore, the initial claim made in the question is incorrect; the given series does not converge. Hence, we cannot find a sum for this series.