The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series
For the perimeter P (a, b) of an ellipse with the semi-axes a ≥ b ≥ 0 a sequence Qn(a, b) is constructed such that the relative error of the approximation P(a,b) ≈ Qn(a,b) satisfies the following inequalities
0 ≤ −(P(a,b)−Qn(a,b))/P(a,b) ≤ (1 −q2)n+1/(2n+1)2
≤ (1/(2n+1)2) e-q^2(n+1),
true for n ∈ N and q = b/a ∈ [0,1].
 G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608.
 B. W. Barnard, K. Pearce and L. Schovanec, Inequalities for the perimeter of an Ellipse, J. Math. Anal. Appl. 260 (2001), 295–306.
 C.-P. Chen and F. Qi, Best upper and lower bounds in Wallis’ inequality, Journal of the Indonesian Mathematical Society 11 (2005), no. 2, 137–141.
 C.-P. Chen and F. Qi, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc. 133(2005), 397–401.
 C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma functions and proof of Wallis’ inequality, Tamkang Journal of Mathematics 36 (2005), no. 4, 303–307.
 V. G. Cristea, A direct approach for proving Wallis’ ratio estimates and an improvement of Zhang-Xu-Situ inequality, Studia Univ. Babes,-Bolyai Math. 60 (2015), 201–209.
 J.-E. Deng, T. Ban and C.-P. Chen;Sharp inequalities and asymptotic expansion associated with the Wallis sequence, J. Inequal. Appl., (2015), 2015:186.
 S. Dumitrescu, Estimates for the ratio of gamma functions using higher order roots, Studia Univ. Babes,-Bolyai Math. 60 (2015), 173–181.
 S. Guo, J.-G. Xu and F. Qi,Some exact constants for the approximation of the quantity in the Wallis’ formula, J. Inequal. Appl., (2013), 2013:67.
 S. Guo, Q. Feng, Y.-Q. Bi and Q.-M. Luo, A sharp two-sided inequality for bounding the Wallis ratio, J. Inequal. Appl., (2015), 2015:43.
 B.-N. Guo and Feng Qi, On the Wallis formula, International Journal of Analysis and Appli- cations 8 (2015), no. 1, 30–38.
 J. Ivory, A new series for the rectification of the ellipsis; together with some observations on the evolution of the formula a2 + b2 − 2ab cos φn, Trans. Royal Soc. Edinburgh 4 (1796), 177–190.
 A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 (2012), 833-837.
 V. Lampret, The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations, Math. Mag. 74 (2001), No. 2, pp. 109–122.
 V. Lampret, Wallis’ Sequence Estimated Accurately Using an Alternating Series, J. Number. Theory. 172 (2017), 256-269.
 V. Lampret, A Simple Asymptotic Estimate of Wallis Ratio Using Stirlings Factorial Formula, Bull. Malays. Math. Sci. Soc. (2018), doi.org/10.1007/s40840-018-0654-5.
 C. E. Linderholm and A. C. Segal, An Overlooked Series for the Elliptic Perimeter, Mathe- matics Magazine, 68(1995)3, 216–220.
 C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Math. Soc. Simon Stevin, 17 (2010), pp. 929–936.
 C. Mortici, New approximation formulas for evaluating the ratio of gamma functions, Math. Comput. Modelling 52 (2010), pp. 425–433.
 C. Mortici, A new method for establishing and proving new bounds for the Wallis ratio, Math. Inequal. Appl. 13 (2010), 803–815.
 C. Mortici, Completely monotone functions and the Wallis ratio, Appl. Math. Lett. 25 (2012), 717–722.
 C. Mortici and V. G. Cristea, Estimates for Wallis’ ratio and related functions, Indian J. Pure Appl. Math. 47 (2016), 437–447.
 C. A. Maclaurin, A treatise of fluctions in two books, Vol.2, T. W. and T. Ruddimans, Edin- burgh 1742.
 F. Qi and C. Mortici, Some best approximation formulas and the inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368.
 , F. Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza–Issues of Analysis 7 (25) (2018), no. 1, 104–115.
 J.-S. Sun and C.-M. Qu, Alternative proof of the best bounds of Wallis’ inequality, Commun. Math. Anal. 2 (2007), 23–27.
 M. B. Villarino, A Direct Proof of Landen’s Transformation, arXiv:math/0507108v1 [math.CA].
 M. B. Villarino, Ramanujan’s inverse elliptic arc approximation, Ramanujan J., 34 (2014), no. 2, 157–161.
 S. Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., 1988–2009.
 X.-M. Zhang, T. Q. Xu and L. B. Situ Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007) 1, art. 17, 9 p.