The expression given is:

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

and the heading suggests applying the generalized ratio test to determine convergence.

Step-by-step analysis:

1. Generalized Ratio Test: For a series $\sum a_n$, the ratio test uses: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ If $L < 1$, the series converges absolutely. If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive.

   Here, $a_n = (-1)^n \frac{2^n x^n}{n^3}$.

2. Calculate the ratio: $\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1} \frac{2^{n+1} x^{n+1}}{(n+1)^3}}{(-1)^n \frac{2^n x^n}{n^3}}$ Simplify: $\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1}}{(-1)^n} \cdot \frac{2^{n+1} x^{n+1}}{(n+1)^3} \cdot \frac{n^3}{2^n x^n}$ Since $\frac{(-1)^{n+1}}{(-1)^n} = -1$, we get: $\frac{a_{n+1}}{a_n} = - \frac{2 \cdot 2^n x \cdot x^n}{(n+1)^3} \cdot \frac{n^3}{2^n x^n}$ Simplify further: $\frac{a_{n+1}}{a_n} = - \frac{2x \cdot n^3}{(n+1)^3}$

3. Take the limit as $n \to \infty$: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{2x \cdot n^3}{(n+1)^3} \right|$ Expand $(n+1)^3$ using the binomial theorem: $(n+1)^3 = n^3 + 3n^2 + 3n + 1$ So: $\frac{n^3}{(n+1)^3} = \frac{n^3}{n^3 (1 + \frac{3}{n} + \frac{3}{n^2} + \frac{1}{n^3})} \to \frac{1}{1} = 1 \quad \text{as } n \to \infty.$

   Therefore: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 2|x|$

4. Convergence Condition: For absolute convergence: $2|x| < 1 \implies |x| < \frac{1}{2}$

Conclusion:

The series converges absolutely for $|x| < \frac{1}{2}$.

Would you like a more detailed breakdown of any step, or have additional questions? Here are five related questions to explore:

1. What happens if $|x| = \frac{1}{2}$?
2. How does the alternating nature of the series affect the convergence test?
3. Can we apply other tests, such as the root test, to analyze this series?
4. What are the implications of absolute versus conditional convergence?
5. How would the series behave if $n^3$ in the denominator were replaced with $n^2$?

Tip:

When working with convergence tests, ensure you simplify ratio expressions fully before taking limits.