To determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^n(6n+12)}{4n+7}$ is convergent or divergent, let's analyze the terms step-by-step.

Step 1: Determine the Type of Series

The series has a factor of $(-1)^n$, which suggests it is an alternating series.

Step 2: Apply the Alternating Series Test (Leibniz’s Test)

The Alternating Series Test requires two conditions for convergence:

1. The terms $a_n = \left|\frac{6n+12}{4n+7}\right|$ should be decreasing.
2. $\lim_{n \to \infty} a_n = 0$.

Let’s check each condition:

Condition 1: Monotonic Decrease

To determine if $a_n = \frac{6n+12}{4n+7}$ is decreasing, we could take the derivative with respect to $n$ and check if it’s negative, or check manually by comparing successive terms. Given the form, as $n$ grows larger, the fraction generally decreases because the growth of the numerator and denominator are similar, but the denominator slightly outpaces the numerator as $n \to \infty$.

Condition 2: Limit of Terms

$\lim_{n \to \infty} \frac{6n + 12}{4n + 7} = \lim_{n \to \infty} \frac{6 + \frac{12}{n}}{4 + \frac{7}{n}} = \frac{6}{4} = \frac{3}{2} \neq 0.$ Since the limit does not approach zero, this means that the Alternating Series Test fails.

Step 3: Apply the Divergence Test

If $\lim_{n \to \infty} a_n \neq 0$, the series diverges by the Divergence Test.

Conclusion

The series diverges.

Would you like further details on any step or have specific questions about alternative methods for testing this series?

Here are five questions to expand your understanding:

1. Why does the Divergence Test require the limit of terms to be zero for possible convergence?
2. What is the Alternating Series Test, and why does it work for some series but not others?
3. How would the series behave differently if the numerator’s growth was slower?
4. Could the Ratio Test or Root Test also help analyze this series?
5. What is the difference between absolute and conditional convergence?

Tip: For alternating series, always check if the non-alternating part converges to zero before applying the Alternating Series Test.