Roots of $\sum_{k=0}^n x^k$ How can I go on to show that the roots of $1+x+x^2+x^3+\ldots+x^n$ are exactly
$$\exp\left(\frac{2ki\pi}{n+1}\right)$$
for $k=1,\ldots,n$?

$(1-x)(1+x+x^{2}+...+x^{n})= 1-x^{n+1}$. So the roots are same as the roots of $1-x^{n+1}=0$ except for $x=1$. 
