<s>[INST] Find the fourier series of the function Find the fourier series of the function
$g(x) = \sum\limits_{n=1}^\infty \frac{sin(nx)}{6^n sin(x)}$ for $x \not= k\pi$, and $g(k\pi) = \lim_{x\to k\pi} g(x)$, $(k \in \mathbb{Z})$
 [/INST] $g(x)=\sum\limits_{n=1}^\infty \frac{\alpha^n}{sin(x)}(sin(n-2)x cos2x + cos(n-2)x sin2x)$
$ = \sum\limits_{n=1}^\infty \alpha^n\frac{sin(n-2)x}{sin(x)} + 2\sum\limits_{n=1}^\infty \alpha^n cos(n-1)x$
$= -\alpha + \alpha^2\sum\limits_{n=1}^\infty\alpha^n \frac{sin(nx)}{sinx} + 
2\sum\limits_{n=1}^\infty\alpha^n cos(n-1)x$
where we can find $g(x)$ to be:
$g(x) = \frac{\alpha}{\alpha^2-1} + 2\sum\limits_{n=1}^\infty \frac{\alpha^n}{1-\alpha^2}cos(n-1)x$
$ = \frac{\alpha}{1-\alpha^2} + 2\alpha\sum\limits_{n=1}^\infty\frac{\alpha^n}{1-\alpha^2} cos(nx)$
now in our case $\alpha = \frac{1}{6}$
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