Prove the following DFT for a periodic sinc sequence or ideal lowpass filter
Hints:

1) Definition of the unnormalized DFT:

$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-i\frac{2\pi}{N}kn}  $$

2) Geometric Summation Formula:

$$ \sum_{n=0}^{N-1} r^n = \frac{1-r^N}{1-r} $$

3) Exponential form of Sine:

$$ \sin( \theta ) = \frac{ e^{i\theta} - e^{-i\theta} }{ 2i } $$

4) Definition of unnormalized sinc:

$$ \operatorname{sinc}(x) = \frac{\sin(x)}{x} $$