I am having trouble finding the same answer as the solution manual for this sequence.
The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\ldots N-1\} \\ 0 & \text{for odd } n \in \{0\ldots N-1\} \end{cases} $$
Setting up the DFT equation I agree with the solution manual $$ X[k]=\sum_0^{N-1}x[n]W^{kn}_{N}=\sum_0^{\frac{N}{2}-1}W^{k2n}_{N}=\sum_0^{\frac{N}{2}-1}e^{\frac{-j2\pi k(2n)}{N}} $$
However when apply the closed form geometric series formula $\sum_0^Na^k=\frac{1-a^{N+1}}{1-a}$ I get $$ \frac{1-e^{-j2\pi k}}{1-e^{\frac{-j2\pi k}{N}}} $$
while the solution manual says its
$$\frac{1-e^{-j2\pi k}}{1-e^{\frac{-j\pi k}{N}}}$$
Note their lower exponent does not have a 2. Could someone explain where and how the 2 goes away? Or does someone agree that the provided answer is incorrect?
This is question 8.5c from Oppenheim's Discrete-Time Signal Processing 3e.