To solve or analyze the given infinite series:

$\sum_{n=1}^\infty \frac{1}{9n^2 - 3n - 2},$

Step 1: Factorize the denominator

We first factorize the quadratic expression $9n^2 - 3n - 2$ in the denominator:

$9n^2 - 3n - 2 = (3n - 2)(3n + 1).$

This factorization works because: $(3n - 2)(3n + 1) = 9n^2 + 3n - 6n - 2 = 9n^2 - 3n - 2.$

Thus, the series becomes: $\sum_{n=1}^\infty \frac{1}{(3n - 2)(3n + 1)}.$

Step 2: Partial fraction decomposition

Next, decompose the fraction using partial fractions: $\frac{1}{(3n - 2)(3n + 1)} = \frac{A}{3n - 2} + \frac{B}{3n + 1}.$

To find $A$ and $B$, clear the denominators: $1 = A(3n + 1) + B(3n - 2).$

Expanding and collecting like terms: $1 = (3A + 3B)n + (A - 2B).$

Matching coefficients for $n$ and the constant term:

1. $3A + 3B = 0$ (coefficient of $n$),
2. $A - 2B = 1$ (constant term).

Solve this system:

• From $3A + 3B = 0$, we get $A = -B$.
• Substitute $A = -B$ into $A - 2B = 1$: $-B - 2B = 1 \implies -3B = 1 \implies B = -\frac{1}{3}.$
• Thus, $A = -B = \frac{1}{3}$.

So, the partial fraction decomposition is: $\frac{1}{(3n - 2)(3n + 1)} = \frac{1/3}{3n - 2} - \frac{1/3}{3n + 1}.$

Step 3: Rewrite the series

Substitute the partial fractions into the series: $\sum_{n=1}^\infty \frac{1}{(3n - 2)(3n + 1)} = \frac{1}{3} \sum_{n=1}^\infty \left( \frac{1}{3n - 2} - \frac{1}{3n + 1} \right).$

Step 4: Recognize the telescoping nature

Observe that the terms telescope, meaning most terms cancel out when expanded:

• For $n = 1$: $\frac{1}{3(1) - 2} - \frac{1}{3(1) + 1} = \frac{1}{1} - \frac{1}{4}$,
• For $n = 2$: $\frac{1}{3(2) - 2} - \frac{1}{3(2) + 1} = \frac{1}{4} - \frac{1}{7}$,
• And so on.

The general pattern cancels intermediate terms, leaving only: $\text{Remaining terms} = \frac{1}{3} \left( 1 - \lim_{n \to \infty} \frac{1}{3n + 1} \right).$

As $n \to \infty$, $\frac{1}{3n + 1} \to 0$, so: $\text{Remaining terms} = \frac{1}{3} \cdot 1 = \frac{1}{3}.$

Final Answer:

The value of the series is: $\boxed{\frac{1}{3}}$

Do you want more explanation or another example? Let me know!

Related Questions:

1. What is a telescoping series, and how does it work in general?
2. How can partial fraction decomposition be applied to other series problems?
3. What is the test for convergence of an infinite series?
4. Can we solve this using the comparison test?
5. How do you recognize when a series is telescoping?

Tip: Telescoping series simplify complex problems by canceling out intermediate terms. Always check for factorization to identify such series.