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

Question

Let $u_n = \frac{4\sin\left(n-\frac{1}{2}\right)\sin\frac{1}{2}}{\cos(2n-1)+\cos 1}.$ Using the formulae $\cos P \pm \cos Q$, show that $$u_n = \frac{1}{\cos n} - \frac{1}{\cos(n-1)}.$$[2] Hence find $\displaystyle\sum_{r=1}^{N} u_n$.[2] Explain why the infinite series $u_1 + u_2 + u_3 + \dots$ does not converge.[1]

Accepted Answer

(i) $u_n=\frac1{\cos n}-\frac1{\cos(n-1)}$ (ii) $\sum_{r=1}^{N}u_r=\frac1{\cos N}-1$ (iii) The infinite series does not converge.

2022 TGM P1 Q5 [out of 2025 syllabus]

2022 TGM P1 Q5 [out of 2025 syllabus]

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