- Given that
$$( 1 + x ) ^ { n } = 1 + \sum _ { r = 1 } ^ { \infty } \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { 1 \times 2 \times \ldots \times r } x ^ { r } \quad ( | x | < 1 , x \in \mathbb { R } , n \in \mathbb { R } )$$
- show that
$$( 1 - x ) ^ { - \frac { 1 } { 2 } } = \sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \left( \frac { x } { 4 } \right) ^ { r }$$
- show that \(\left( 9 - 4 x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }\) can be written in the form \(\sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \frac { x ^ { 2 r } } { 3 ^ { q } }\) and give \(q\) in terms of \(r\).
- Find \(\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r } { 9 } \times \left( \frac { x } { 3 } \right) ^ { 2 r - 1 }\)
- Hence find the exact value of
$$\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r \sqrt { } 5 } { 9 } \times \frac { 1 } { 5 ^ { r } }$$
giving your answer as a rational number.