LTI: Is my calculation of the frequency response correct?

Question

i am not sure if my computation is correct. Hope someone can have a look at it. Here is my problem:

\begin{aligned} H[e^{j\omega}] & = \sum_{m=0}^{\infty} (\frac{1}{2} e^{-j\omega})^n = \frac{1}{1-0.5e^{-j\omega}}\\ H[e^{j\omega}] & = \frac{1}{(1.25 - cos{\omega})^{\frac{1}{2}}}\\ \end{aligned}

I got to my solution by looking at a similar example.
But i don't know how they got rid of the sum.

Answer 1

The summation is simply a geometric series. You remove the summation by using the formula

\begin{equation} 1+x+x^2 +\ldots = \frac{1}{1-x} \end{equation}

For the magnitude response, you just have to use your formula $|c|^2 = cc^*$. If $c = a+ib$, this gives you $|c| = \sqrt{a^2 + b^2}$.