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A function $f$ is defined as follow:

$$ f(x)=\sum_{n=1}^{\infty}\frac{b_{n}}{(x-a_{n})^{2}+b_{n}^{2}}\;\;, x\in \mathbb R $$

where $(a_{n}, b_{n})$ are points in the $xy$-plane, $b_{n}>0$ for all $n$. When is the function $f(x)$ bounded away from zero? that is $f(x)\geq a>0$, for some $a>0$, for all $x\in \mathbb R$. I believe that this would somehow depend on the points $(a_{n}, b_{n})$, for example if $\{b_{n}\}$ converges to zero or not, but I cannot find out how!

Thanks

*EDIT: I still don't know when $f(x)$ will be bownded away from zero! *

Edit: $f(x)$ is bounded above by some constant, say $C>0$.

  • 1
    Instead of "strictly positive" you can say: "bounded away from zero".2012-02-20
  • 0
    Yes, thanks! I fixed it.2012-02-20

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