Final answer:
Part (a) converges by the Alternating Series Test, as the terms approach zero and are decreasing. Part (b) appears to contain a typo, and more context is needed to provide an accurate evaluation of convergence or divergence.
Step-by-step explanation:
When asked to determine which of the following series converge, we are dealing with the area of mathematical analysis specifically series convergence tests. The question provided two parts, labeled (a) and (b), each requiring a different method to determine convergence. For part (a):
Σ (-1)^(k+1) * k / (k^2 + 6) looks like an alternating series. We apply the Alternating Series Test. In this test, we need to check if the absolute value of the terms is decreasing and approaches zero as k approaches infinity. Using the limit comparison with 1/k, which is a p-series with p=1 that diverges, we see that as k goes to infinity, the terms of the given series approach zero and are decreasing because k^2+6 grows faster than k. Therefore, by the Alternating Series Test, the series converges. For part (b): It seems there's been a misunderstanding in the question format with lim (n→n) -n² + 6. Presumably, the student was asked to determine the limit of a function as n approaches infinity, a key concept in establishing series convergence. However, a limit like lim (n→n) is not meaningful. It's likely that the question was meant to ask for the limit as n approaches infinity. In this case, the limit of -n² + 6 as n approaches infinity is negative infinity, which suggests divergence; however, without the correct context, we can't be certain of the intended series to test for convergence/divergence.