Using the definition of Fourier coefficients in a Fourier series representation, I have managed to show the following series:
$$\cos \alpha x= \frac {\sin \alpha \pi}{\alpha \pi}+ \frac{2 \alpha}{\pi}\sin \alpha \pi \sum_{k=1}^{\infty}(-1)^{k-1}\frac{\cos kx}{k^2-\alpha^2}$$
However, I now need to use l'Hopital's Rule to determine the limit of the RHS of this series as $\alpha$ tends to $n$ where $n$ is a positive integer.
I have heard about this rule in basic analysis but not really studied it properly before, could someone just clarify exactly how it works and how I am using it to find the limit here?
Hint: For fixed integer $n$, we can separate the summation on RHS: $$\frac {\sin \alpha \pi}{\alpha \pi}+ \frac{2 \alpha}{\pi}\sin \alpha \pi \sum_{k=1}^{\infty}(-1)^{k-1}\frac{\cos kx}{k^2-\alpha^2}$$ $$=\frac {\sin \alpha \pi}{\alpha \pi}+ \frac{2 \alpha}{\pi}\sin \alpha \pi (-1)^{n-1}\frac{\cos nx}{n^2-\alpha^2}+ \frac{2 \alpha}{\pi}\sin \alpha \pi \sum_{k=1,k\neq n}^{\infty}(-1)^{k-1}\frac{\cos kx}{k^2-\alpha^2}$$ $$=\frac {\sin \alpha \pi}{\alpha \pi}+ \frac{2(-1)^{n-1}\cos nx}{\pi} \dfrac{\alpha\sin \alpha \pi}{n^2-\alpha^2}+ \frac{2 \alpha}{\pi}\sin \alpha \pi \sum_{k=1,k\neq n}^{\infty}(-1)^{k-1}\frac{\cos kx}{k^2-\alpha^2}$$ Now take the limit as $\alpha\to n$ with l'Hopital's Rule.