2 By considering the identity $\cos[(2n-1)\alpha]-\cos[(2n+1)\alpha]\equiv2\sin\alpha\sin2n\alpha,$ show that if $\alpha$ is not an integer multiple of $\pi$ then $\sum_{n=1}^N \sin (2n\alpha) = \tfrac12 \cot\alpha -\tfrac12 \operatorname{cosec}\alpha\cos[(2N+1)\alpha]. \tag{4}$ Deduce that the infinite series $\sum_{n=1}^\infty \sin (\tfrac23n\pi) \tag1$ does not converge.
Hello. I bring into your consideration the following question on series. The first part of the question is relatively straightforward, using the method of differences on the left side we then divide by the constant term $2\sin\alpha$ to obtain what is required in the first part. However, I am completely nonplussed as to what train of thought/logic has to be followed for the second part. Any input or clarification will be much appreciated. Do keep in mind though that this question holds only the weight of a single mark so will probably not require exhaustive calculation/elaborate proof.