The task involves determining whether the given series converge absolutely, conditionally, or diverge. Let's analyze the two series one by one.

Problem (a):

$\sum_{n=1}^\infty \frac{(-1)^n}{(3n-2)!}$

Analysis:

1. Absolute Convergence:
   To check for absolute convergence, consider the series of absolute values: $\sum_{n=1}^\infty \frac{1}{(3n-2)!}.$ The factorial $(3n-2)!$ grows extremely quickly, and we know that series of the form $\sum \frac{1}{n!}$ converge because the terms tend to zero extremely fast. Thus, this series converges absolutely.

2. Conditional Convergence:
   Since the series already converges absolutely, it automatically converges conditionally as well.

Problem (b):

$\sum_{n=1}^\infty (-1)^n \frac{n(3n^2 - 1)}{5 + 2n^2}.$

Analysis:

1. Absolute Convergence:
   Consider the series of absolute values: $\sum_{n=1}^\infty \left| \frac{n(3n^2 - 1)}{5 + 2n^2} \right|.$ For large $n$, the numerator behaves like $3n^3$ (dominant term), and the denominator behaves like $2n^2$, so the terms simplify asymptotically to: $\frac{n(3n^2)}{2n^2} \sim \frac{3n}{2}.$ Since $\frac{3n}{2}$ grows without bound, the terms do not tend to zero, and the series diverges absolutely.

2. Conditional Convergence:
   Now consider the original alternating series. For alternating series to converge, the terms must:

   • Decrease monotonically ($a_{n+1} < a_n$).
   • Tend to zero ($\lim_{n \to \infty} a_n = 0$).

   Since the absolute value of the terms grows like $\frac{3n}{2}$, they do not tend to zero. Therefore, the series does not converge conditionally either.

Final Results:

• (a): The series converges absolutely.
• (b): The series diverges.

Do you want a step-by-step explanation for any specific part? Let me know!

Related Questions:

1. How does the factorial growth in series (a) ensure convergence?
2. Why does the alternating series test fail in part (b)?
3. Can series with terms tending to infinity ever converge conditionally?
4. How can we test for divergence in non-alternating series?
5. What are the differences between absolute and conditional convergence?

Tip:

When working with factorial-based terms, always compare their growth rate to standard tests (e.g., ratio test, comparison with $e^n$). This simplifies deciding convergence behavior.