This image contains a series of tasks related to series convergence and divergence, specifically applying Leibniz's test and evaluating limits of terms in alternating series. Let's go through each part:

a) Series:

$\sum_{k=1}^{\infty} \frac{(-1)^k}{3k + 2}$

1. Positiveness of terms: The series contains alternating terms, meaning the signs alternate. We don't need to confirm whether terms are positive because the alternating factor is clear with $(-1)^k$.

2. Testing for convergence: Leibniz's test applies to alternating series, and it requires that:

   • $a_k \to 0$ as $k \to \infty$.
   • $a_k$ should be decreasing for all $k$.

   Now, $a_k = \frac{1}{3k + 2}$ Evaluating the limit: $\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{3k + 2} = 0$ Since this condition is met, we can conclude that the series converges.

3. Conclusion: The series converges.

b) Series:

$\sum_{k=1}^{\infty} (-1)^k k^5$

1. Positiveness of terms: This is an alternating series, but we should check whether the terms go to zero. Here, the general term is $a_k = k^5$, which grows as $k$ increases, and it does not approach zero.

   Therefore, Leibniz's test cannot be applied because $\lim_{k \to \infty} a_k \neq 0$.

2. Conclusion: The series diverges.

c) Series:

$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k - 1}$

1. Positiveness of terms: The series alternates, and again we are dealing with an alternating series. The term is of the form $a_k = \frac{1}{2k - 1}$.

2. Testing the limit of the terms: $\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{2k - 1} = 0$ So the condition for Leibniz's test is met. The terms go to zero.

3. Conclusion: This series converges by Leibniz's test.

Let me know if you need further explanation on any part, or if you'd like me to go over any steps in detail. Here are a few follow-up questions to expand your understanding:

1. How do we handle cases where the general term $a_k$ does not approach zero?
2. What other convergence tests are useful for series besides Leibniz's test?
3. How does the rate of decrease of $a_k$ influence the convergence of alternating series?
4. Can Leibniz’s test be applied to series with non-alternating terms?
5. What is the importance of checking the behavior of $a_k$ as $k \to \infty$ for determining convergence?

Tip: Always verify if the terms of the series decrease and approach zero before applying convergence tests.