I would want to ask what is the z-transform of this signal x[k]:
$$x[k]=\sum_{n=0}^{\infty}c^n * \delta [k-2n] \:; c<0 $$
I know that I have to use the geometric series but I really don't know where to even start because of that sum over $n$.
$$X(z)=\sum_{k=0}^{\infty}(\sum_{n=0}^{\infty}c^n * \delta [k-2n])*Z^{-k} $$
The z-transform of $\delta [k-k_0]$ is $Z^{-k_0}$. But what is going on with the sum form n=0?
EDIT: Could it be:
$$X(z)=1+cZ^{-2}+c^2Z^{-4}+c^3Z^{-6}+..... $$
which makes
$$X(z)=\sum_{m=0}^{\infty}(cZ^{-2})^m = \frac {1}{1-c*Z^{-2}} \:;|cZ^{-2}|<1 $$
Edit2: 