Check convergence and find the sum $\sum_{n=1}^{\infty} \frac{1}{9n^2+3n-2}$ 
$$\sum_{n=1}^{\infty} \frac{1}{9n^2+3n-2}$$

I have starting an overview about series, the book starts with geometric series and emphasizing that for each series there is a corresponding infinite sequence.
For convergence I can look at the partial sums, but how can I find the sum of the series?

Note that
$$ \frac{1}{9n^2 + 3n - 2} = \frac{1}{(3n + 2)(3n - 1)} = \frac{1}{3} \left( \frac{1}{3n - 1} - \frac{1}{3n + 2} \right) $$
for all natural numbers $n$.
The partial sums of the sum that you are interested in are then given by
$$ \sum_{n=1}^N \frac{1}{9n^2 + 3n - 2} = \frac{1}{3} \sum_{n=1}^N \left( \frac{1}{3n - 1} - \frac{1}{3n + 2} \right)  = \frac{1}{3} \sum_{n=1}^N \left( \frac{1}{3n - 1} - \frac{1}{3(n + 1) - 1} \right) $$
One sees that the sum telescopes; that is to say all of the terms cancel except for the first and last, leaving us with
$$ \frac{1}{3} \left( \frac{1}{3 \cdot 1 - 1} - \frac{1}{3(N + 1) - 1} \right) = \frac{1}{6} - \frac{1}{9N + 6} $$
As $N$ tends to $\infty$, this tends to $\frac{1}{6}$, and so we see that the sum converges and that its value is $\frac{1}{6}$.
