To analyze whether the given series converge absolutely, conditionally, or diverge, let’s evaluate each series step by step:

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

1. Absolute Convergence Test:

   • Consider the absolute value of the terms: $\frac{1}{n^3}$.
   • This is a $p$-series with $p = 3 > 1$. Since $p > 1$, the series $\sum_{n=1}^\infty \frac{1}{n^3}$ converges.
   • Hence, the given series converges absolutely.

2. Conclusion:

   • Because the series converges absolutely, it also converges conditionally.

(b) $\sum_{n=1}^\infty \frac{(-1)^n}{n + \log n}$

1. Alternating Series Test:

   • Let $a_n = \frac{1}{n + \log n}$.
   • Check if $a_n$ is positive, decreasing, and approaches zero as $n \to \infty$:
     • $a_n > 0$: True for $n \geq 1$.
     • $a_n$ decreases because $n + \log n$ increases as $n$ grows.
     • $\lim_{n \to \infty} a_n = 0$: True, as the denominator grows without bound.
   • Thus, the series satisfies the conditions of the Alternating Series Test and converges.

2. Absolute Convergence Test:

   • Check the series $\sum_{n=1}^\infty \frac{1}{n + \log n}$ for absolute convergence.
   • Compare it to $\sum_{n=1}^\infty \frac{1}{n}$, which diverges (harmonic series).
   • Since $n + \log n > n$ for $n \geq 1$, $\frac{1}{n + \log n} < \frac{1}{n}$, but the comparison still leads to divergence.
   • Hence, the series does not converge absolutely.

3. Conclusion:

   • The series converges conditionally but not absolutely.

Final Results:

1. (a): Converges absolutely (and conditionally).
2. (b): Converges conditionally but not absolutely.

Do you want additional explanations or examples?
Here are five related questions to expand your understanding:

1. What are the key differences between absolute and conditional convergence?
2. How does the Alternating Series Test guarantee convergence?
3. Why do $p$-series converge only when $p > 1$?
4. Can a conditionally convergent series be rearranged to diverge?
5. How does logarithmic growth affect series convergence?

Tip: When analyzing convergence, always test for absolute convergence first—it simplifies the problem significantly.