2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).
- Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that
$$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
- Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
- Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.