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What can we calculate with a simple calulator?
For example, take $$a_0=x,~b_0=y$$
$$a_1=\frac{a_0+\sqrt{a_0b_0}}{2},~~~~b_1=\frac{b_0+\sqrt{a_0b_0}}{2}$$
$$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$
Then we obtain as a limit the 'logarithmic mean' of $x,y$:
$$a_\infty=b_\infty=\frac{x-y}{\ln x-\ln y}$$ - FixedPoint[(# + Sqrt[#[[1]] #[[2]]])/2 &, {1., 2.}]
{1.4426950408889632, 1.4426950408889636}
$L(1,2)=\frac1{\ln2}$ |
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