This exercise is about determining whether the given series is absolutely convergent, conditionally convergent, or divergent. Here’s a step-by-step breakdown of the solution:

Problem Analysis

The series is: $\sum_{n=1}^{\infty} \frac{(-1)^n (6n)}{\sqrt{25n^4 - 1}}$

It’s an alternating series (indicated by the factor $(-1)^n$), so we'll evaluate both absolute and conditional convergence.

Part 1: Testing $\sum |a_n|$

Absolute Convergence

To determine if the series is absolutely convergent, we consider the absolute value of $a_n$: $|a_n| = \frac{6n}{\sqrt{25n^4 - 1}}$

We use the Limit Comparison Test with a simpler comparison series of the form $\frac{c}{n^p}$.

1. Choice of Comparison Series: The leading term in the denominator, $\sqrt{25n^4} = 5n^2$, suggests comparing $|a_n|$ with $\frac{6}{5n}$.

2. Calculations:

   • The constant $c = \frac{6}{5}$ and $p = 1$.
   • Using the limit comparison, we find: $\lim_{n \to \infty} \frac{|a_n|}{b_n} = 1$
   • Since $\sum b_n = \sum \frac{c}{n}$ diverges (harmonic series), $\sum |a_n|$ also diverges.

This confirms that the series is not absolutely convergent.

Part 2: Conclusion on Convergence

Since $\sum |a_n|$ diverges, we now check if the series $\sum a_n$ converges conditionally.

Conditional Convergence

The series $\sum a_n$ is an alternating series, and we can apply the Alternating Series Test (Leibniz’s test).

1. Conditions of the Alternating Series Test:
   • Monotonicity: $\{b_n\} = \frac{6n}{\sqrt{25n^4 - 1}}$ is ultimately decreasing for $n > 6/5$.
   • Limit Condition: $\lim_{n \to \infty} b_n = 0$.

Since both conditions are satisfied, the series $\sum a_n$ converges conditionally.

Final Answer

• The series $\sum a_n$ is conditionally convergent.

Would you like further details on any specific part of this solution? Here are related questions that might be useful:

1. How does the Limit Comparison Test work in determining convergence?
2. Why does the harmonic series diverge, and how is it applied in comparison?
3. What is the Alternating Series Test, and why is it useful here?
4. How do we determine if a series is absolutely convergent?
5. What are examples of other conditionally convergent series?

Tip: For alternating series, always verify if the terms decrease and approach zero, as this is critical for conditional convergence.