On the identity $M(a,b)I(a,b)=\frac{\pi}{2}$ Given two positive real numbers $a$ and $b$, one can define their arithmetic-geometric mean as $M(a,b):=\ell$, where
$$ (\ell,\ell)=\lim_{k\to\infty}(a_k,b_k)\quad\text{and}\quad\begin{cases}a_{k+1}=\frac{a_k+b_k}{2} \\ b_{k+1}=\sqrt{a_kb_k} \\ a_0=a,\ b_0=b.\end{cases} $$
On the other hand, one can define the elliptic integral
$$ I(a,b):=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}}. $$
According to this interesting blog post by Paramanand, the famous formula
$$\boxed{M(a,b)I(a,b)=\frac{\pi}{2}}$$
was discovered by Gauss, thanks to some (actually very hidden) numerical evidence. Using a clever change of variables, it can be shown that $I(a,b)=I(\frac{a+b}{2},\sqrt{ab})$, from which the identity follows.
Actually, Gauss' first proof was a rather tedious computation with power series.

Is there a simple, natural proof of the identity, explaining how one
  could guess it (without numerical evidence) in the first place?

 A: 
Is there a simple, natural proof of the identity, explaining how one could guess it (without numerical evidence) in the first place?

IF you are asking for the intuition behind the formally proven identity, then consider the following:


*

*We are all familiar with the formulae expressing the circle's area and circumference in terms of its one and only radius.

*An ellipse is a “circle” with two (unequal) “radii”.

*This being said, let us try and “guess”, as you've put it, the formulae for the elliptic area and “circumference”.

*It stands to reason, then, that $A=\pi~r^2=\pi~r~r$ should generalize into $A=\pi~r_1~r_2,$ since products of lengths yield areas.

*But what about $L=2\pi~r$ ? Should the $d=2r$ be interpreted as $d=r+r,$ yielding a possible generalization of $L=\pi~(r_1+r_2)$ ? Or, on the contrary, should the $2$ be coupled with $\pi,$ and the stand-alone r be regarded as a “mean” elliptical radius, ultimately related to an integral, which concept is intimately connected to that of area, which would then, in turn, lend credence to using $r=\sqrt{r_1~r_2}~$ here as well, as opposed to $r=\dfrac{r_1+r_2}2$ ? As it happens, ellipses themselves, and not just the people studying them, seem unable to make up their little elliptic minds on this issue, and, in their confusion (or perhaps out of a politically correct desire to reconcile both parts and offend none), they have opted neither for the former $r=AM(r_1,r_2),$ nor for the latter $r=GM(r_1,r_2),$ but for something in between, namely $r=AGM(r_1,r_2).$
