Originally Posted by

**Deadstar** For this question I have to use the Riemann-Lebesgue Lemma to prove that...

$\displaystyle \lim_{N \rightarrow \infty} \int_{-\pi}^{\pi} \Bigg{(} \frac{1}{\sin(x/2)} - \frac{2}{x} \Bigg{)} \sin((N + 1/2)x) dx = 0$.

With the Riemann-Lebesgue Lemma lemma being...

If $\displaystyle g$ is an integrable function on $\displaystyle [-\pi, \pi]$, then $\displaystyle \lim_{|n| \rightarrow \infty} G(n) = 0.$.

Where $\displaystyle G(n) = \frac{-1}{2\pi} \int_{-\pi}^{\pi} g(x - (\pi/n))e^{-inx} dx$

Now I can do this question easily by expanding the brackets but I'm unsure as to how to apply the lemma as clearly expanding the brackets never uses

that...