Dirichlet kernel

**Deadstar** · March 25th 2010, 10:29 AM

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

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

**CaptainBlack** · March 26th 2010, 02:31 AM

Originally Posted by Deadstar

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

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