Gravity center of a discrete signal
Your double sums do not make sense. The numerator and the denominator of the expression for the center of gravity are numbers, and it's pointless to take the discrete-time Fourier transform (DTFT) of these numbers. What you are supposed to do is express the center of gravity in terms of the DTFT of $x[n]$.

Since this is a homework problem, instead of solving it for you I'll give you a few hints that should help you figure out the solution by yourself:


The discrete-time Fourier transform (DTFT) of the sequence $nx[n]$ is given by $$\text{DTFT}\{nx[n]\}=j\frac{dX(\omega)}{d\omega}$$
The sum over a sequence (if it exists) is equal to the DTFT of that sequence evaluated at $\omega=0$: $$\sum_{n=-\infty}^{\infty}x[n]=X(0)$$