Evaluate $${\lim_{x \rightarrow 0} \frac{1-\cos x\cos 2x\cdots \cos nx}{x^2}}$$
It should be $$\frac{1}{12}n(n+1)(2n+1)$$ but I don't know how to prove that. I am also aware that $\lim\limits_{\theta \to 0} \dfrac{1-\cos \theta}{\theta ^2}=\dfrac{1}{2}$, but I don't know how to use that.
Best Answer
Use $$1-\prod_{k=1}^n\cos{kx}=1-\prod_{k=1}^n\left(1-2\sin^2\frac{kx}{2}\right)\sim2\sum_{k=1}^n\left(\frac{kx}{2}\right)^2= \frac{x^2}{2}\sum_{k=1}^nk^2$$