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To determine if the series

\[
\sum_{n=1}^{\infty} \frac{3}{(3n - 2)(3n + 1)}
\]

converges or diverges and to find its sum if it converges, we can simplify and evaluate this series by applying partial fraction decomposition.

### Step 1: Partial Fraction Decomposition

We can rewrite the general term:

\[
\frac{3}{(3n - 2)(3n + 1)} = \frac{A}{3n - 2} + \frac{B}{3n + 1}
\]

Multiplying both sides by \((3n - 2)(3n + 1)\), we get:

\[
3 = A(3n + 1) + B(3n - 2)
\]

Expanding and collecting terms in terms of \(n\):

\[
3 = (3A + 3B)n + (A - 2B)
\]

By equating coefficients, we get two equations:

1. \(3A + 3B = 0\)
2. \(A - 2B = 3\)

Solving these equations, we find \(A = 1\) and \(B = -1\). Therefore:

\[
\frac{3}{(3n - 2)(3n + 1)} = \frac{1}{3n - 2} - \frac{1}{3n + 1}
\]

### Step 2: Rewrite the Series Using the Partial Fractions

The series can now be written as:

\[
\sum_{n=1}^{\infty} \left( \frac{1}{3n - 2} - \frac{1}{3n + 1} \right)
\]

### Step 3: Recognize the Telescoping Nature of the Series

Writing out the first few terms of this series:

\[
\left( \frac{1}{1} - \frac{1}{4} \right) + \left( \frac{1}{4} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{10} \right) + \cdots
\]

Notice that most terms cancel out, leaving only:

\[
1 - \lim_{k \to \infty} \frac{1}{3k + 1}
\]

### Step 4: Evaluate the Sum

As \(k \to \infty\), \(\frac{1}{3k + 1} \to 0\). Thus, the sum of the series is:

\[
S = 1
\]

### Conclusion

The series converges, and its sum is \(1\).

Would you like further details on any of these steps, or do you have questions on related concepts?

Here are some additional questions related to this topic:
1. What are the conditions for a series to be classified as telescoping?
2. How does partial fraction decomposition assist in evaluating series?
3. What are other methods for determining convergence of a series?
4. How does the limit of the partial sum contribute to finding the sum of a converging series?
5. Could this series converge if it were not telescoping?

**Tip:** For telescoping series, always look to simplify terms through partial fractions; it often reveals the cancellations necessary for summing the series.

Use the definition of infinite series to determine if the series converges or diverges. If the series converges, find its sum.

Learn how to determine the convergence and sum of the infinite telescoping series \( \sum_{n=1}^{\infty} \frac{3}{(3n - 2)(3n + 1)} \) by applying partial fraction decomposition. This solution covers the definition of infinite series, partial fraction techniques, and telescoping sums for calculating limits.

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