Finding a DTFT of a signal
Note that \$(-1)^n=e^{i\pi n}\$, so you get

$$\sum_{n=-\infty}^{\infty}x[n](-1)^ne^{-in\theta}=
\sum_{n=-\infty}^{\infty}x[n]e^{i\pi n}e^{-in\theta}=
\sum_{n=-\infty}^{\infty}x[n]e^{-in(\theta-\pi)}=X(\theta-\pi)$$

The spectrum is just shifted by \$\pi\$. This is basically a consequence of the modulation property.