2 Let $u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }$.\\
(i) Using the formulae for $\cos P \pm \cos Q$ given in the List of Formulae MF10, show that

$$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$

(ii) Use the method of differences to find $\sum _ { n = 1 } ^ { N } u _ { n }$.\\

(iii) Explain why the infinite series $u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$ does not converge.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q2}}