7 In this question you must show detailed reasoning.
- Show that
$$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$
where \(a\), \(b\) and \(c\) are integers whose values are to be determined.
You are given that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\) exists and is equal to \(\frac { 1 } { 6 } \pi ^ { 2 }\).
- Show that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }\) exists and is equal to \(( \pi - 1 ) ( \pi + 1 )\).