Prove that $\frac{1}{a_1} + \frac{2}{a_1+a_2} + \frac{3}{a_1+a_2+a_3}<2(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}) $ Prove that $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + \frac{3}{a_1+a_2+a_3}<2(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}) $$ where $a_1, a_2, a_3 >0$.
From AM-HM I got that $\frac{2}{a_1+a_2}\le \frac{1}{2}(\frac{1}{a_1}+\frac{1}{a_2})$ and $\frac{3}{a_1+a_2+a_3}\le \frac{1}{3}(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3})$, but adding these is not enough.
 A: Further to my comment and your progress with AM-HM
$$\frac{1}{a_1}<\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$$
$$\frac{2}{a_1+a_2}\leq \frac{1}{2}\left(\frac{1}{a_1}+\frac{1}{a_2}\right)<\frac{1}{2}\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)$$
$$\frac{3}{a_1+a_2+a_3}\leq \frac{1}{3}\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)$$
Add together
$$\frac{1}{a_1}+\frac{2}{a_1+a_2}+\frac{3}{a_1+a_2+a_3}<\color{red}{\left(1+\frac{1}{2}+\frac{1}{3}\right)}\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\right)<\color{red}{2}\cdot ...$$
A: Adding the inequalities given by you:
$$\dfrac{1}{a_1}+\dfrac{2}{a_1+a_2}+\dfrac{3}{a_1+a_2+a_3}<\dfrac{1+\frac{1}{2}+\frac{1}{3}}{a_1}+\dfrac{\frac{1}{2}+\frac{1}{3}}{a_2}+\dfrac{\frac{1}{3}}{a_3}$$
$$\dfrac{1+\frac{1}{2}+\frac{1}{3}}{a_1}+\dfrac{\frac{1}{2}+\frac{1}{3}}{a_2}+\dfrac{\frac{1}{3}}{a_3}<\dfrac{2}{a_1}+\dfrac{2}{a_2}+\dfrac{2}{a_3}$$
$\therefore \dfrac{1}{a_1}+\dfrac{2}{a_1+a_2}+\dfrac{3}{a_1+a_2+a_3}<\dfrac{2}{a_1}+\dfrac{2}{a_2}+\dfrac{2}{a_3}$
