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.

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.