Deriving Ramanujan's Ellipse approximation

See the Gauss-Kummer formula in section 7 of the following paper

http://web.tecnico.ulisboa.pt/~mcasquilho/compute/com/,ellips/PerimeterOfEllipse.pdf
i.e. 
\[P=\pi(a+b) \left[ 1+(\frac{1}{2})^2h^2+(\frac{1}{2.4})^2h^4+(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8})^2h^8+ \dots\right],\]
where $h=\frac{a-b}{a+b}$.

On the other if one computes the taylor series expansion of the function 
\[f(h)=\pi (a + b) \left(1 + \frac{3h^2}{10 + \sqrt{4 - 3h^2}}\right)\]
near $h=0$ we get 
\[f(h)=\pi(a+b) \left[ 1+(\frac{1}{2})^2h^2+(\frac{1}{2.4})^2h^4+(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8})^2h^8+ \dots\right].\]
Ineed $f(h)$, Ramanujan approximation for circumference of an ellipse, is an $O(h^{10})$ approximation of the circumference.