2022 TGM P1 Q5

Let \(u_n = \frac{4\sin\left(n-\frac{1}{2}\right)\sin\frac{1}{2}}{\cos(2n-1)+\cos 1}.\)

1. Using the formulae \(\cos P \pm \cos Q\), show that \[u_n = \frac{1}{\cos n} - \frac{1}{\cos(n-1)}.\][2]
2. Hence find \(\displaystyle\sum_{r=1}^{N} u_n\).[2]
3. Explain why the infinite series \(u_1 + u_2 + u_3 + \dots\) does not converge.[1]