To solve this problem, we need to apply Cauchy's root test ($Признак Коши$) to determine if the given series converges absolutely, conditionally, or diverges. Here's the step-by-step explanation:

1. Cauchy's Root Test Formula:

For a series $\sum a_n$, the test involves calculating:

$\limsup_{n \to \infty} \sqrt[n]{|a_n|}$

• If the limit is $< 1$: The series converges absolutely.
• If the limit is $= 1$: The test is inconclusive.
• If the limit is $> 1$: The series diverges.

Here, the general term of the series is:

$a_n = (-1)^{n-1} \left(\frac{3n + 1}{4n + 7}\right)^{2n}$

Since Cauchy's test depends only on the magnitude, we consider:

$|a_n| = \left(\frac{3n + 1}{4n + 7}\right)^{2n}$

2. Applying the Root Test:

To simplify:

$\sqrt[n]{|a_n|} = \left(\frac{3n + 1}{4n + 7}\right)^2$

Now calculate the limit as $n \to \infty$:

$\lim_{n \to \infty} \left(\frac{3n + 1}{4n + 7}\right)^2$

Simplify the fraction inside the parentheses:

$\frac{3n + 1}{4n + 7} = \frac{3 + \frac{1}{n}}{4 + \frac{7}{n}} \to \frac{3}{4} \text{ as } n \to \infty$

So:

$\lim_{n \to \infty} \left(\frac{3n + 1}{4n + 7}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$

3. Conclusion:

Since $\frac{9}{16} < 1$, the series converges absolutely. According to the instructions, we input 1 to indicate absolute convergence.

Would you like me to clarify any part or discuss how the absolute value affects convergence? 😊

Related Questions:

1. What is the difference between absolute and conditional convergence?
2. How does Cauchy's root test compare to other convergence tests like the ratio test?
3. What happens if the limit in the root test equals 1?
4. How do alternating signs in the series influence convergence?
5. Can we apply other tests to verify convergence for this series?

Tip:

Always simplify the inner term before applying limits—it can save a lot of time in convergence tests!