\sum\limits_{n \geqslant 2} {\frac{1}
{{n^2 - 1}}} = \sum\limits_{n \geqslant 2} {\left( {\frac{1}
{2}\frac{1}
{{n - 1}} - \frac{1}
{2}\frac{1}
{{n + 1}}} \right)} = \frac{1}
{2}\left( {1 + \frac{1}
{2}} \right) + \sum\limits_{n \geqslant 2} {\left( {\frac{1}
{2}\frac{1}
{{n + 1}} - \frac{1}
{2}\frac{1}
{{n + 1}}} \right)} = \frac{3}
{4}