$Squeeze$ Ellipse and Calculus $\Longrightarrow Ellipsulus$

$S_{n}=lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{k}{n^{2}+n+2k}$

$S_{n}=\frac{1}{n^{2}+n+2}+\frac{2}{n^{2}+n+4}+...+\frac{n}{n^{2}+n+2k}$

$\frac{1}{n^{2}+3n}+\frac{2}{n^{2}+3n}+...+\frac{n}{n^{2}+3n}<s_{n}<\frac{1}{n^{2}+n+2}(1+2+....+n)$

$\frac{1}{2}\leq lim_{n\rightarrow\infty}\leq\frac{1}{2}$

By squeeze theorem

$lim_{n\rightarrow\infty}=\frac{1}{2}$

clearly we have from the properties of ellipses

$\frac{SM}{S'M'}=\frac{S'P}{SP}$

$d=\pi$

$S=\pi ab$

$\therefore\frac{S}{d}=ab$

Let the co-ordinates of the point $K_{r}$ be $(1+Rcos\theta,1+Rsin\theta)$ here $R=AK_{r}$

$\Rightarrow\frac{R^{2}sin^{2}\theta}{b^{2}}+\frac{R^{2}cos^{2}\theta}{a^{2}}=1$

we can clearly observe that

$lim_{n\rightarrow\infty}\frac{1}{n}\sum_{r=1}^{n}AK_{r}^{2}=lim_{n\rightarrow\infty}\frac{1}{n}\sum_{r=1}^{n}\frac{1}{(\frac{cos^{2}(\frac{r\pi}{2n})}{a^{2}}+\frac{sin^{2}(\frac{r\pi}{2n})}{b^{2}})}=\int_{0}^{1}\frac{a^{2}b^{2}}{b^{2}cos^{2}\frac{\pi x}{2}+a^{2}sin^{2}\frac{\pi x}{2}}dx=ab$

we can further also prove that this limit exists only for c=1

2c=2