The purpose of the question is to find out how H (harmonic mean) departs from A (Arithmetic mean).
$A= \frac{x1+x2+x3}{3}$ and $H= \frac{3}{1/x1+1/x2+1/x3}$
To that end, assume that: $x1=A+e1,x2=A+e2, x3=A+e3$, and write a Taylor expansion to order 2 of: $1/x1$ and $1/x2$ and $1/x3$. then the expansion of H with respect to e1, e2, e3 supposed to be small quantities.
Show that, $A-H \stackrel{}{\approx} \frac{e1^2+e2^2+e3^2}{3A}$ + terms of higher order
I have tried expanding the taylor series of second order and inserted them in H and then proceeded to A-H but the form requested still did not apear, I am not sure what to do.
Let $x_1=x_2-\Delta$, $x_3=x_2+\Delta$ $$A=x_2 \qquad \text{and}\qquad H=x_2 \left(1+\frac{2 \Delta ^2}{\Delta ^2-3 x_2^2}\right)$$
$$A-H=\frac{2 \Delta ^2 x_2}{3 x_2^2-\Delta ^2}\sim \frac23{\frac{\Delta ^2} {x_2}}=\frac23{\frac{\Delta ^2} {A}}$$