I'm attending this course (Coursera: Audio Signal Processing for Music Applications) in which the professor derives a general equation for Discrete Fourier Transform (DFT) for a complex sinusoid. The following is the screenshot of the slide he used:
While the derivation is totally fine, I'm having trouble understanding the concluding statement
if $k \neq k_0, denominator \neq 0$ & $numerator = 0$ thus $ X_1[k]=N$ for $k=k_0$ & $X_1[k]=0$ for $k \neq k_0$
I did try value substitution and all, but all that can't seem to justify the statement. My understanding is that if $k=k_0$, both the numerator and the denominator would be zero and there would be no way the result is $N$ and I have no clue about the first part of the statement either. Or I'm missing something here (or forgotten some basic school math here). What's going on here?