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December 12th, 2018, 01:26 PM   #1
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Richardson extrapolation

Consider a regular n-sided polygon tangential to and enclosing the unit circle to approximate $\pi$.
a) Use geometrical arguments to show that the half-perimeter of the polygon $C_n=n\tan(\pi /n) \geq \pi$.
b) Calculate $C_4$ and $C_8$.
c) Use Richardson extrapolation with $h = 1/n^2$ to improve upon your results.
For part b) I've got:
\[ C_4 = 4\tan(\frac{\pi}{4}) = 4\times 1 = 4, \]
\[ C_8 = 8\tan(\frac{\pi}{8}) = 8 \frac{\sin(\frac{\pi}{8})}{\cos(\frac{\pi}{8})} = 8\frac{\frac{\sqrt{2-\sqrt{2}}}{2}}{\frac{\sqrt{2+\sqrt{2}}}{2}} = 8\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\]
By drawing, I have been able to do the part a) but I have no clue how to do part c). Can someone provide me some help? Thank you.

Last edited by skipjack; December 12th, 2018 at 07:00 PM.
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December 12th, 2018, 01:29 PM   #2
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December 12th, 2018, 07:04 PM   #3
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tan($\pi$/8) = √2 - 1
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