# Sum of each difference between the area of a unit circle and the area of the $n$-th regular polygon inscribed in it

A regular polygon (with $$n$$ sides) inscribed in a unit circle ($$R=1$$) has the area: $$A_n=\frac{1}{2}n\sin\frac{2\pi}{n}$$.

So, the difference between the area of the unit circle ($$\pi$$) and that of the $$n$$-th regular polygon equals: $$D_n=\pi-\frac{1}{2}n\sin\frac{2\pi}{n}$$.

If we sum all these differences, we get: $$\sum_{n=3}^\infty D_n=\sum_{n=3}^\infty \pi-\frac{1}{2}n\sin\frac{2\pi}{n}$$ Does this sum converge?

Yes it converges. Let $$E_n$$ be the difference between the area of the inscribed and the area of the outscribed regular polygon (with $$n$$ sides). Clearly $$E_n>D_n$$ for all $$n$$.
$$E_n=n\cdot \frac{\sin^3\frac\pi n}{\cos\frac\pi n}< n \cdot\frac{\frac{\pi^3}{n^3}}{\cos\frac\pi 3}=\frac{\pi^3}{\cos\frac\pi 3}\cdot\frac1{n^2}$$
So we have $$D_n<\frac{\pi^3}{\cos\frac\pi 3}\cdot\frac1{n^2}$$.
$$\sum_{n=3}^\infty\frac{\pi^3}{\cos\frac\pi 3}\cdot\frac1{n^2}$$ converges, so $$\sum_{n=3}^\infty D_n$$ converges too.