Fourier Series: Deriving the Dirichlet Kernel

From my Wavelets course (study guide for exam)
PROBLEM
Derive the formula for the Dirichlet kernel.

SOLUTION
\begin{eqnarray*}
D_n(t)&=& \sum^n_{-n} e^{ikt}=e^{-int}\sum_0^{2n} e^{ijt}, j:=k+n\\
&=& e^{-int}\left(\displaystyle\frac{e^{i(2n+1)t-1}}{e^{it}-1}\right)\\

&=& e^{-int}\left(\displaystyle\frac{e^{i(2n+1)t/2}}{e^{it/2}}\right)\cdot \left(\displaystyle\frac{e^{i(2n+1)t/2}-e^{i(2n+1)t/2}}{e^{it/2}-e^{-it/2}}\right)\\
&=& \left(\displaystyle\frac{\sin(n+1/2)t}{\sin(t/2)}\right)

\end{eqnarray*}

Note that the above is for $0<|t| \leq \pi$. If $t=0$, $D_n(t)=2n+1$.