1. ## Dirichlet kernel

How would you show that $\sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right )}{\sin(x/2)}$

WITHOUT using the method shown here...
Dirichlet kernel - Wikipedia, the free encyclopedia

Alternatively how would you prove that

$\sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}$?

How would you show that $\sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right )}{\sin(x/2)}$

WITHOUT using the method shown here...
Dirichlet kernel - Wikipedia, the free encyclopedia

Alternatively how would you prove that

$\sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}$?
The second is trivial (and hence so is the first), multiply by $r^n$ and you have $\sum_{k=1}^{2n} r^k$ which is well known, but this is what the Wikipedia article does.

CB