How to find average and power of signal expressed by fourier series

Question

I need to find the average and the power of this signal:

$$x(n)=\sum_{k=1}^{\infty}2^{-k}e^{j2{\pi}kn}$$

The problem is that the summation starts at 1 and not at 0, and a part of that how can I find the average of a signal expressed in this way?

Answer 1

HINT:

Note that we have

$$\begin{align} x(n)&=\sum_{k=1}^\infty 2^{-k}e^{i2\pi kn}\\\\ &=\sum_{k=1}^\infty\left(\frac{e^{i2\pi n}}{2}\right)^k\\\\ &=\frac{e^{i2\pi n}}{2-e^{i2\pi n}}\\\\ &=\frac{2e^{i2\pi n}-1}{5-8\cos(2\pi n)} \end{align}$$

Can you proceed now?