<s>[INST] Series Convergence of Harmonic Means 
Let $\{x_n\}$ be a sequence of real numbers such that $0< x_1 <x_2$. If $$x_n=  \frac{2}{\frac{1}{x_{n-1}}+\frac{1}{x_{n-2}}}, $$then show that 
  $$\lim_{n\to\infty}x_n=\frac{3x_1x_2}{2x_1+x_2}.$$

 [/INST] Note that
\begin{align}
& x_n = \frac{2}{\frac{1}{x_{n-1}} + \frac{1}{x_{n-2}}} \\
\iff & \frac{1}{x_{n-1}} + \frac{1}{x_{n-2}} = \frac{2}{x_{n}} \\
\iff & a_n = \frac{1}{2}a_{n-1} + \frac{1}{2}a_{n-2},
\end{align}
where $a_i = \frac{1}{x_i}$.
It's easy to verify that $$a_n = \alpha + \beta\left(-\frac{1}{2}\right)^n.$$
Consequently, $$\lim_{n\to \infty}a_n = \alpha.$$
Using $$a_1 = \frac{1}{x_1} = \alpha-\frac{\beta}{2},$$
and $$a_2 = \frac{1}{x_2} = \alpha+\frac{\beta}{4},$$
one gets $$\lim_{n\to\infty}x_n = \frac{1}{\alpha} = \frac{3x_1 x_2}{2x_1 + x_2}.$$
</s>