On The Perimeter of an Ellipse

Let E be the ellipse with major and minor radii a and b respectively, and Pbe its perimeter, then P = lim ۴ tan(p/n)(a + b + ۲) Σ a۲ cos۲ (۲k-۲)Pi/n+ sin۲ (۲k-۲)Pi/n; where n = ۲m. So without considering the limit, it gives a reasonable approxi-mation for P, it means that we can choose n large enough such that the amountof error be less than any given small number. On the other hand, the formulasatises both limit status b→a and b→۰ which give respectively P = ۲a andP = ۴a.