$a_{n+1}=\frac{a_n+b_n}{2}$
$b_{n+1}=\sqrt{a_{n+1}b_{n}}$
also, $a_1=a$ and $b_1=b$
evaluate $$\lim_{n\to\infty}a_{n}$$ This question is from guillaume musso's novel La Jeune Fille et la Nuit . I remember that the answer included $\arccos$ thing..I asked this question few days ago but didn't get satisfactory answer. Please help me.
Hint. Assuming that $a < b$, find $c$ and $\theta$ such that
$$ \frac{1}{\tan\theta} = \frac{a}{c}, \qquad \frac{1}{\sin\theta} = \frac{b}{c}. $$
Then the solution of the sequence is given by
$$ a_n = \frac{c}{2^{n-1}\tan(\theta/2^{n-1})}, \qquad b_n = \frac{c}{2^{n-1}\sin(\theta/2^{n-1})}, $$
and hence the common limit is given by $c/\theta$. This sequence can be traced back to Archimedes' algorithm to approximating the circumference of a circle by inscribed and circumscribed polygons, see this.