<s>[INST] Evaluate $\sum\limits_{n=0}^{\infty} \frac{\cos(nx)}{2^n}$ where $\cos x = \frac15$ Evaluate 
$$\sum_{n=0}^{\infty} \dfrac{\cos(nx)}{2^n}$$
where $\cos x = \frac{1}{5}$.
This is a complex number question. But I don’t know where to start. Maybe need to use the DeMoivre’s Theorem?
 [/INST] If $S=\sum_{r=0}^\infty y^r\cos r x$
Using scale of relation from this or $\#187$ of this(downloadable)
$$(1-2y\cos x+y^2)S$$
$$=1+y\cos x+y^2\cos2x+y^3\cos3x+\cdots$$
$$-2y\cos x-2y^2\cos^2x-2y^3\cos2x\cos x-2y^4\cos3x\cos x-\cdots$$
$$+y^2+y^3\cos x+y^4\cos2x+y^5\cos3x+\cdots$$
$$=1+y(-\cos x)+y^2(\cos2x-2\cos^2x+1)+y^3(\cos3x-2\cos3x\cos x+\cos x)+\cdots$$
$$\implies S=\dfrac{1-y\cos x}{1-2y\cos x+y^2}$$
Here $2y=1$
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