Is $\lim\limits_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots + \cos{\frac{n\pi}{n}} \right)$ a Riemann sum? This is probably simple, but I'm solving a practice problem:
$\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots +\cos{\frac{n\pi}{n}} \right)$
I recognize this as the Riemann sum from 0 to $\pi$ on $\cos{x}$, i.e. I think its the integral
$\int_0^\pi{ \cos{x}dx }$
which is 0, but the book I'm using says it should be
$ \frac{1}{\pi}\int_0^\pi{ \cos{x}dx }$
Still 0 anyway, but where did the $\frac{1}{\pi}$ in front come from?

The key to this last assertion is the simple fact that $$\cos(\pi - x) = -\cos(x).$$  Said symmetry can be observed directly from the definition of the cosine function via the unit circle.
