I need to know if this series converges and if so, to which value. The series is given by:
\begin{equation} M_2(X) = \lim_{N \to \infty} \frac{1}{12m}{\sum_{i = 1}^{N}{\frac{(x_{i}^2 + x_{i}x_{{i+1}} + x_{{i+1}}^2)(x_{i}y_{{i+1}} - x_{{i+1}}y_{i})}{\pi r^2}}} \end{equation}
I thought to transform it in polar coordinates $$x = c_1 + r\cos(2\pi i/N)$$ $$ x = c_2 + r\sin(2\pi i/N)$$ Where ($c_1, c_2$) is a center and $i$ and $N$ are natural numbers. This equation is from a polygon with $N$ sides.
In polar co-ordinates the equation above is
$$ \lim_{N \to \infty}\frac{1}{\pi r^2} \sum_{i = 1}^{N} ((\alpha^2 +\alpha\beta +\beta^2)(\alpha\gamma -\beta\gamma)) $$
Where $\alpha=(c_1+r\cos(\frac{2\pi i}{N}))$ $\beta=(c_1+r\cos(\frac{2\pi (i+1)}{N}))$ $\gamma=(c_2+r\sin(\frac{2\pi (i+1)}{N}))$