# The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series

@article{Lampret2019ThePO, title={The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series}, author={Vito Lampret}, journal={Cubo (Temuco)}, year={2019} }

For the perimeter \(P(a,b)\) of an ellipse with the semi-axes \(a\ge b\ge 0\) a sequence \(Q_n(a,b)\) is constructed such that the relative error of the approximation \(P(a,b)\approx Q_n(a,b)\) satisfies the following inequalities
\(0\le -\frac{P(a,b)-Q_n(a,b)}{P(a,b)}\le\frac{(1-q^2)^{n+1}}{(2n+1)^2}\)
\(\le \frac{1}{(2n+1)^2}\,e^{-q^2(n+1)},\)
true for \(n\in{\mathbb N}\) and \(q=\frac{b}{a}\in[0,1]\).

## 2 Citations

Basic asymptotic estimates for powers of Wallis’ ratios

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## References

SHOWING 1-10 OF 30 REFERENCES

Inequalities for the perimeter of an ellipse

- MathematicsThe Mathematical Gazette
- 2014

The perimeter of the ellipse x 2/a 2 + y 2/b 2 = 1 is 4J (a, b), where J (a, b) is the ‘elliptic integral’ This integral is interesting in its own right, quite apart from its application to the…

COMPLETELY MONOTONIC FUNCTION ASSOCIATED WITH THE GAMMA FUNCTIONS AND PROOF OF WALLIS' INEQUALITY

- Mathematics
- 2005

Abstract. We prove: (i) A logarithmically completely monotonic function is completely mono-tonic. (ii) For x > 0 and n = 0,1,2,..., then(−1) n ln p xΓ(x)x + 1/4Γ(x + 1/2) ! (n) > 0.(iii) For all…

An Elloquent formula for the perimeter of an ellipse

- Mathematics
- 2012

T he values of complete elliptic integrals of the first and the second kind are expressible via power series representations of the hypergeometric function (with corresponding arguments). The…

Ramanujan’s inverse elliptic arc approximation

- Mathematics
- 2005

We suggest a continued fraction origin to Ramanujan’s approximation to $(\frac{a-b}{a+b})^{2}$ in terms of the arc length of an ellipse with semiaxes a and b.

A Simple Asymptotic Estimate of Wallis’ Ratio Using Stirling’s Factorial Formula

- MathematicsBulletin of the Malaysian Mathematical Sciences Society
- 2018

Several new, accurate, simple, asymptotic estimates of Wallis’ ratio $$w_n:=\prod \nolimits _{k=1}^{n} \frac{2k-1}{2k}$$wn:=∏k=1n2k-12k are obtained on the bare the Bernoulli coefasis of Stirling’s…

Estimates for Wallis’ ratio and related functions

- Mathematics
- 2016

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice…

A direct approach for proving Wallis ratio estimates and an improvement of Zhang-Xu-Situ inequality

- Mathematics
- 2015

In time, inequalities about Wallis ratio and related functions were presented by many mathematicians. In this paper, we show how estimates on the Wallis ratio can be obtained using the asymptotic…

Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, p, and the Ladies Diary

- Mathematics
- 1988

Paper 8: Gert Almkvist and Bruce Berndt, “Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi, and the Ladies Diary,” American Mathematical Monthly, vol. 95 (1988), pg. 585–608.…

The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations

- Computer Science
- 2001

The existence of the calculator might suggest that computational leverage provided by calculus is no longer needed, but the ability to compute is still immeasurably enriched by the power of calculus.