Question: I'm supposed to determine whether this sum diverges or converges and if it converges then find its value: 
$$
\sum\_{n=2}^\infty \frac{1}{n^2-1}.
$$

Using the comparison test I eventually showed that this converges. But I can't figure out how to show what this sum converges to. The only sums we actually found values for in my notes are geometric series which this clearly isn't.

I saw that I could use partial fraction decomposition to represent the terms as 
$$\frac{1/2}{n-1}- \frac{1/2}{n+1} $$
but that just gets me $\infty - \infty$, so this isn't the way to do it.

I'm not sure how to find the value of this sum. I don't need the full solution but a hint would be appreciated. Thanks. :)

Answer: 
Use partial fraction decomposition to represent the sum and notice that it is a telescoping series.
Since $$\frac{1}{n^2-1}=\frac{1}{(n-1)(n+1)}=\frac12\left(\frac{1}{n-1}-\frac{1}{n+1}\right)$$ as you already have, then write for $N\in \mathbb N$  

\begin{align}\sum\_{n=2}^{N}\left(\frac{1}{n-1}-\frac{1}{n+1}\right)&=\left(1-\frac13\right)+\left(\frac12-\frac14\right)+\left(\frac13-\frac15\right)+\dots+\left(\frac1{N-1}-\frac1{N+1}\right)\\[0.2cm]&=1+\frac12+\left(\frac13-\frac13\right)+\dots+\left(\frac1{N-1}-\frac1{N-1}\right)-\frac1N-\frac1{N+1}\\[0.2cm]&=\frac32-\frac1N-\frac1{N+1}\end{align} In other words, this sum *telescopes.* Now let $N\to \infty$ (and of course do not forget 1/2 in front of the sum) to conclude that $$\sum\_{n=2}^{+\infty}\frac{1}{n^2-1}=\lim\_{N\to+\infty}
\frac12\sum\_{n=2}^N\left(\frac{1}{n-1}-\frac1{n+1}\right)=\frac12\lim\_{N\to+\infty}\left(\frac32-\frac1N-\frac1{N+1}\right)=\frac34$$