<s>[INST] Infinite inequality with logarithm Solve inequality
$3 - \log_{0.5}x - \log^2_{0.5}x - \log^3_{0.5}x - \cdots \ge 4\log_{0.5}x$
Any suggestions how to start?
 [/INST] We have
$$\begin{align}
3-\log_{0.5}x-\log^2_{0.5}x-\log^3_{0.5}x-\cdots&=4-\sum_{n=0}^{\infty}\log^n_{0.5}x\\\\
&=4-\frac{1}{1-\log_{0.5}x}\\\\
&=\frac{3-4\log_{0.5}x}{1-\log_{0.5}x}
\end{align}$$
for $|\log_{0.5}x|<1\implies 0.5<x<2$.  Now note that the inequality 
$$\frac{3-4\log_{0.5}x}{1-\log_{0.5}x}\ge 4\log_{0.5}x$$
is equivalent to 
$$\left(2\log_{0.5}x-3\right)\left(2\log_{0.5}x-1\right)\ge 0$$
which implies $\log_{0.5}x\le 0.5$ or $x\ge \sqrt{2}/2$.  Therefore, we have 
$$\bbox[5px,border:2px solid #C0A000]{\sqrt{2}/2\le x<2}$$
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