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A family of high order approximations of Ramanujan type for perimeter of an ellipse. (English) Zbl 1467.33021

Summary: Inspired by Ramanujan’s unusual approximations for the perimeter of an ellipse with semiaxis 1 and \(r\in (0,1)\), a family of approximations of Ramanujan type is constructed. We find the sharp lower and upper approximations have very high accuracy, and better than some known ones.

MSC:

33E05 Elliptic functions and integrals
26A48 Monotonic functions, generalizations
26D15 Inequalities for sums, series and integrals
40A25 Approximation to limiting values (summation of series, etc.)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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