Fourier transform of $ne^{-an}u[n]$
You're right, the first Fourier transform correspondence in your reference is wrong. It should be

$$\mathcal{F}\{e^{-\alpha n}u[n]\}=\frac{1}{1-e^{-\alpha}e^{-j\omega}}\tag{1}$$

You just need to substitute $a=e^{-\alpha}$ and use the formulas you know.