Q:

Find the common limit of $\frac{2}{1/a_n+1/b_n}$ and $\sqrt{a_n b_n}$

Let $a_0=1$ and $b_0=2$, then
\begin{align}
a_{n+1} &= \frac{2}{\frac{1}{a_n}+\frac{1}{b_n}}, \\
b_{n+1} &= \sqrt{a_n b_n}.
\end{align}
The sequences $(a_n)$ and $(b_n)$ converge to the same limit.
Is it possible to find that common limit?
I know $(a_n)$ is increasing and $(b_n)$ is decreasing.
I can show that they converge to the same limit, but I don't see what can be this limit.
Is using a computer the only way to find this limit?
Is there a theoretical method to find it?

A:

The limit is called the geometric-harmonic mean. Check http://en.wikipedia.org/wiki/Geometric%E2%80%93harmonic_mean

