As is known to us, the limit of the sequence $\{a_n\},\{b_n\}\ $where$\ a_0=a,b_0=b,a_{n+1}=\dfrac{a_n+b_n}{2},b_{n+1}=\sqrt{a_n b_n}\ $ is related to the Arithmetic–geometric mean of $a,b$, which can be expressed as below
$AG(a,b)=\dfrac{\pi}{2I(a,b)}$, where $\displaystyle I(a,b)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}}$
Now I wonder whether we can calculate the limit of the following sequence
$\ a_0=a,b_0=b,a_{n+1}=\dfrac{a_n+b_n}{2},b_{n+1}=\dfrac{a_n-b_n}{\ln a_n-\ln b_n}$
(without the loss of generality we can assume $a>b>0$)
and I'd like to call this limit the Arithmetic–logarithmic mean
Anyone can solve this?