$1\mathrm{.}1$ The value of $$ can be estimated from the calculations of the side lengths of regular polygons inscribing a circle. In general,

${}_k={}_{}+\frac{c_1}{k}+\frac{c_2}{k^2}+\frac{c_3}{k^3}+$cdots,

where ${}_k$ is the ratio of the perimeter to the diameter of a regular $k-$sided polygon. Determine the approximate value of ${}_{}$ from ${}_8=3\mathrm{.}061467,{}_{16}=3\mathrm{.}121445,{}_{32}=3\mathrm{.}136548,$ and ${}_{64}=3\mathrm{.}140331$ of the inscribing polygons. Which $c_i$ is most significant and why? What happens if we use the values from the polygons circumscribing the circle, for example, ${}_8=3\mathrm{.}313708,{}_{16}=3\mathrm{.}182598,{}_{32}=3\mathrm{.}151725,$ and ${}_{64}=3\mathrm{.}144118?$