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Conditions are determined under which 3F2 ( n; a; b; a + b +2 ;" n + 1; 1) is a monotone function of n satisfying ab 3F2 ( n; a; b; a + b +2 ;" n +1 ; 1) ab 2F1 (a; b; a + b +2 ; 1): Motivated by a conjecture of M. Vuorinen, the corollary that 3F2 n; 1 2 ; 1 2 ;1 ;" n +1 ; 1 4 ,f or 1 > 1 4 and n 2; is used to determine surprising hierarchical relationships among the thirteen known historical approximations of the arc length of an ellipse. This complete list of inequalities compares the Maclaurin series coecients of 2F1 with the coecients of each of the known approximations, for which maximum errors can then be established. These approximations range over four centuries from Kepler's in 1609 to Almkvist's in 1985 and include two from Ramanujan.

A Monotonicity Property Involving 3F2 and Comparisons of the Classical Approximations of Elliptical Arc Length

A Monotonicity Property Involving ₃F₂ and Comparisons of the Classical Approximations of Elliptical Arc Length