Show that
$$\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { A } { r ^ { 2 } - 1 }$$
where \(A\) is a constant to be found.
7
Hence use the method of differences to show that
$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } \equiv \frac { a n ^ { 2 } + b n + c } { 4 n ( n + 1 ) }$$
where \(a\), \(b\) and \(c\) are integers to be found.