[ Trigonometric Identities ]

Complex numbers help solve the problem $\quad]$

Find the limit $\lim _{n \rightarrow \infty}\left(1+\frac{1}{2} \cos x+\ldots+\frac{1}{2^{n}} \cos n x\right)$.
If $z=1 / 2(\cos x+i \sin x)$, then the expression in parentheses is equal to $\operatorname{Re}\left(1+z+z^{2}+\ldots+z^{n}\right)$.

## Answer

$\frac{2(2-\cos x)}{5-4 \cos x}$.