To determine whether this series converges or diverges, let's examine the form and behavior of the terms. The series presented is:

$\sum_{n=1}^{\infty} \frac{7}{2n + 6} (-1)^{n+1}$

where each term is of the form $a_n = \frac{7}{2n + 6}$ with an alternating sign.

Step-by-Step Analysis

1. Identify the Form of the Series: The series is alternating, as indicated by the signs. Thus, we can use the Alternating Series Test (also known as the Leibniz Test) to check for convergence.

2. Alternating Series Test Requirements: The Alternating Series Test states that an alternating series $\sum (-1)^{n+1} a_n$ converges if:

   • $a_n$ is positive for all $n$.
   • $a_n$ is monotonically decreasing.
   • $\lim_{n \to \infty} a_n = 0$.

3. Check Each Condition:

   • Positivity: $a_n = \frac{7}{2n + 6}$ is positive for all $n$.
   • Monotonic Decrease: As $n$ increases, the denominator $2n + 6$ increases, making $a_n$ decrease. Thus, $a_n$ is monotonically decreasing.
   • Limit of $a_n$: $\lim_{n \to \infty} \frac{7}{2n + 6} = 0$

   Since all conditions are met, the series converges by the Alternating Series Test.

Conclusion

The given series converges by the Alternating Series Test.

Would you like more details on this process, or do you have any specific questions?

Here are some related questions you might find interesting:

1. How does the Alternating Series Test differ from other convergence tests?
2. What is the importance of monotonic decrease in the Alternating Series Test?
3. How would you determine divergence if the series didn't alternate in sign?
4. Could this series be tested for absolute convergence, and if so, what would the result be?
5. How does this series compare to other harmonic series in terms of convergence?

Tip: When dealing with alternating series, always verify if the sequence of terms satisfies the conditions of the Alternating Series Test before exploring other methods.