TITLE: Finding the limit using Euler's Formula
QUESTION [5 upvotes]: I need to find the limit of the following, by using Euler's Formula.
$$\lim_{n \to \infty} \left( 1 + \frac{1}{2} \cos{x} + \frac{1}{2^2} \cos{2x}. . . . + \frac{1}{2^n} \cos {nx}\right)$$
Thanks

REPLY [3 votes]: $$
\begin{align}
\sum_{k=0}^\infty\frac{\cos(kx)}{2^k}
&=\frac12\sum_{k=0}^\infty\frac{e^{ikx}+e^{-ikx}}{2^k}\\
&=\frac12\left(\frac1{1-\frac{e^{ix}}2}+\frac1{1-\frac{e^{-ix}}2}\right)\\
&=\frac12\frac{2-\cos(x)}{\frac54-\cos(x)}\\
&=\frac{4-2\cos(x)}{5-4\cos(x)}
\end{align}
$$