othogonality of chebychev polynomials Let the chebyshev polynomials be defined as :

with zeros :

My goal is to show that the family of polynomials :

are orthogonal with respect to 
where :

To achieve this we show :
 

However there is something wrong with the proof, the last expression
fails for m odd and doesn't yield 0 as expected.
Could anyone help me spot the mistake and how to correct it ?

The formula $\sum_{k=0}^{n} x^{k} = \frac{1 - x^{n+1}}{1-x}$ holds for all $x \neq 1$. That means that the argument given only holds for $m$ such that $e^{im\pi/(n+1)} \neq 1$. For an explicit counterexample to the claim following "It is enough to show that..." choose $n=0$, $m=2$. Then the sum is equal to $\cos(\pi) = -1$.
However, with $m \in \{1,...,2n\}$ (and not just for any $m \in \mathbb{Z}^*$ as claimed) we always have $e^{im\pi/(n+1)} \neq 1$, and $\pi r 8$'s answer proves the orthogonality. 
