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.