The problem is to determine the convergence or divergence of the given series:

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

If it converges, calculate the sum.

Step 1: Analyze the series for convergence

The series can be simplified using partial fraction decomposition. Write the general term:

$\frac{2}{(n-1)(n+1)} = \frac{A}{n-1} + \frac{B}{n+1}$

Solve for $A$ and $B$:

$\frac{2}{(n-1)(n+1)} = \frac{A(n+1) + B(n-1)}{(n-1)(n+1)}$

Expanding the numerator:

$2 = A(n+1) + B(n-1) = An + A + Bn - B = (A+B)n + (A-B)$

Equating coefficients:

1. $A + B = 0$
2. $A - B = 2$

From $A + B = 0$, $B = -A$. Substituting into the second equation:

$A - (-A) = 2 \quad \implies \quad 2A = 2 \quad \implies \quad A = 1, \, B = -1$

Thus, the decomposition is:

$\frac{2}{(n-1)(n+1)} = \frac{1}{n-1} - \frac{1}{n+1}$

Step 2: Rewrite the series

The series becomes:

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

This is a telescoping series. Expanding the first few terms:

$\left(\frac{1}{1} - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \cdots$

Most terms cancel out, leaving:

$\text{Sum} = 1 + \frac{1}{2} - \frac{1}{n} - \frac{1}{n+1} \quad \text{as } n \to \infty$

Step 3: Determine convergence and calculate the sum

As $n \to \infty$, $-\frac{1}{n} - \frac{1}{n+1} \to 0$. Thus, the series converges, and the sum is:

$S = 1 + \frac{1}{2} = \frac{3}{2}$

Final Answer:

The series converges, and its sum is:

$\boxed{\frac{3}{2}}$

Would you like further clarification or details? Here are related questions to explore:

1. What is the significance of partial fraction decomposition in series analysis?
2. How do telescoping series simplify the summation process?
3. Can the same method be applied to other similar series? Give examples.
4. What are other tests for convergence besides telescoping?
5. How would the sum change if the starting index were different?

Tip: Always check for telescoping behavior in series involving rational functions; it often simplifies calculations dramatically!