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Let's go through the given series and determine their convergence or divergence.

### (a) \( \sum_{n=1}^{\infty} \frac{(-1)^{n(n-1)/2}}{2^n} \)
This is a geometric series with alternating signs due to \( (-1)^{n(n-1)/2} \). The general form of a geometric series is \( \sum_{n=1}^{\infty} ar^n \), and it converges if \( |r| < 1 \).

In this case, the absolute value of each term is \( \frac{1}{2^n} \), which resembles a geometric series with \( r = \frac{1}{2} \), a constant ratio. Since \( |r| = \frac{1}{2} < 1 \), the series **converges**. 

To check for absolute convergence, we look at the absolute value of the terms:
\[
\sum_{n=1}^{\infty} \frac{1}{2^n}
\]
This is a geometric series with \( r = \frac{1}{2} \), which converges. Therefore, the series **converges absolutely**.

### (b) \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n\sqrt{n}} \)
This is an alternating series. We can apply the **Alternating Series Test (Leibniz's Test)**:
1. The terms \( \frac{1}{n\sqrt{n}} \) are positive, decreasing, and tend to 0 as \( n \to \infty \).
2. \( \lim_{n \to \infty} \frac{1}{n\sqrt{n}} = 0 \).

Thus, the series satisfies the conditions for the alternating series test and **converges conditionally**. 

To test for absolute convergence, consider the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). This is a **p-series** with \( p = 3/2 > 1 \), so it converges. Therefore, the series **converges absolutely**.

### (c) \( \sum_{n=1}^{\infty} \frac{(-1)^n n^2}{n^2 + 1} \)
This is another alternating series, so we apply the **Alternating Series Test**. First, we examine the positive terms:
\[
a_n = \frac{n^2}{n^2 + 1}
\]
As \( n \to \infty \), the terms do not tend to 0, because:
\[
\lim_{n \to \infty} \frac{n^2}{n^2 + 1} = 1
\]
Since the limit is not zero, the series **diverges**.

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Would you like further details or have any specific questions about these results? Here are some related questions:

1. How is the Alternating Series Test applied?
2. What is absolute convergence, and why is it important?
3. How can we identify a geometric series from a given series?
4. Can we always apply the ratio test for alternating series?
5. What is the difference between conditional and absolute convergence?

**Tip:** When working with alternating series, always check if the terms decrease and approach zero to apply the Alternating Series Test effectively.

Determine the convergence or divergence of the given series. In case of convergence, determine whether the series converges absolutely or conditionally.

Determine the convergence or divergence of series, focusing on absolute and conditional convergence. This problem involves geometric series, alternating series, and p-series, suitable for college-level mathematics students.

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