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 \(a\ge b\ge 0\) 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 \(q=\frac{b}{a}\in[0,1]\).