Repeated squared factors in denominator

1. Given that

$$\frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } \equiv \frac { A } { n ^ { 2 } } + \frac { B } { ( n + 1 ) ^ { 2 } }$$

1. determine the value of \(A\) and the value of \(B\)
2. Hence show that, for \(n \geqslant 5\) $$\sum _ { r = 5 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } + a n + b } { c ( n + 1 ) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined.

2 Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find

1. the sum to \(N\) terms,
2. the sum to infinity.