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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.

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

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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.

Determine whether the series \( \sum_{n=1}^{\infty} \frac{(-1)^n(6n+12)}{4n+7} \) is convergent or divergent.

Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n(6n+12)}{4n+7} \) using the Alternating Series Test and Divergence Test. This problem involves key calculus concepts such as the limit of series terms, conditions for the Alternating Series Test, and the Divergence Test.

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