By first expressing \(\frac { 1 } { r ^ { 2 } - 1 }\) in partial fractions, show that
$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } = \frac { 3 } { 4 } - \frac { a n + b } { 2 n ( n + 1 ) }$$
where \(a\) and \(b\) are integers to be found.
Deduce the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ^ { 2 } - 1 }\).
Find the limit, as \(n \rightarrow \infty\), of \(\sum _ { r = n + 1 } ^ { 2 n } \frac { n } { r ^ { 2 } - 1 }\).