Ellipse arc length with series expansion

Question

I have searched in Internet and found many sites, but all these sites give the whole perimeter (circumference) length of an ellipse with series expansion, so I calculate till the accuracy that I want (as in image below). But I would like to calculate the length of an elliptic arc with series expansions till wanted accuracy, I can not deduce so formula.

Answer 1

Plenty of accurate approximations are known for the ellipse perimeter.
If we denote the $p$-th power mean through $$ M_p(a,b) = \sqrt[p]{\frac{a^p+b^p}{2}} \tag{1}$$ due to the results of Muirhead, Alzer and Qiu, the perimeter $L(a,b)$ of an ellipse with semi-axis $a$ and $b$ fulfills

$$ M_{\alpha}(a,b) \leq \frac{L(a,b)}{2\pi}\leq M_{\beta}(a,b),\qquad \alpha=\frac{3}{2},\;\beta=\frac{\log 2}{\log(\pi/2)}.\tag{2} $$