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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?

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Yes; use the Direct Comparison Test to see that it converges indeed.

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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.

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