To evaluate the infinite series

$\sum_{n=2}^{\infty} \frac{(-1)^n (14n)}{2n^2 - 1},$

we can start by analyzing its convergence and then check if it is possible to simplify or evaluate it further.

Step 1: Checking for Convergence

The series has an alternating term, $(-1)^n$, which suggests that the Alternating Series Test (Leibniz’s Test) might be applicable. This test requires that:

1. The absolute value of each term decreases as $n$ increases.
2. The limit of each term as $n \to \infty$ goes to zero.

The term inside the summation, excluding $(-1)^n$, is:

$\frac{14n}{2n^2 - 1}.$

Let's analyze this term to see if it meets these criteria.

Step 2: Simplify the Term $\frac{14n}{2n^2 - 1}$

We can express this term as:

$\frac{14n}{2n^2 - 1} = \frac{14n}{n^2 \left( 2 - \frac{1}{n^2} \right)} = \frac{14}{2n - \frac{1}{n}}.$

As $n \to \infty$, this term behaves like $\frac{14}{2n} = \frac{7}{n}$, which approaches zero. This suggests that the terms do decrease to zero as $n \to \infty$.

Since both conditions of the Alternating Series Test are satisfied, the series is convergent.

Step 3: Attempting to Sum the Series

This series does not have an immediately obvious closed-form expression. However, based on convergence, we can conclude:

$\sum_{n=2}^{\infty} \frac{(-1)^n (14n)}{2n^2 - 1} \text{ converges}.$

Final Answer

The series converges, but it may not simplify to a known closed-form result without further advanced techniques or numerical approximation.

Would you like a deeper explanation of convergence tests, or any specific part of this solution?

Here are some related questions to explore this topic further:

1. What is the Alternating Series Test, and how is it applied?
2. How do we determine the behavior of series terms as $n \to \infty$?
3. Are there other convergence tests we could apply here, such as the Ratio Test or Root Test?
4. What are the properties of alternating series and their convergence criteria?
5. Can this series be approximated numerically for practical applications?

Tip: For alternating series, always check if the terms decrease in absolute value and approach zero—this helps to apply the Alternating Series Test easily.