<s>[INST] Inequality $\sum\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum \frac{1}{x + n^2} $ 
$x\geq0$, then, we have

$$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2} $$


The problem is not easy, even $x=1$. Any help will be appreciated
 [/INST] From a purely algebraic point of view, this problem is quite interesting if we first notice that $$\frac {d}{dx} \Big( \frac {1}{x+n^2} \Big)=-\frac{1}{\left(x+n^2\right)^2}$$ The second point is to recognize that
$$ \sum_{n=1}^{\infty} \frac{1}{x + n^2}=\frac{\pi  \sqrt{x} \coth \left(\pi  \sqrt{x}\right)-1}{2 x}$$ So $$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}-\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2}=\frac{\pi ^2 x \text{csch}^2\left(\pi  \sqrt{x}\right)-1}{4 x}$$ which is always negative.
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