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Function defined by infinite series
A function $f$ is defined as follow:
$ f(x)=\sum_{n\in \mathbb Z} \frac{b_{n}}{(x-a_{n})^{2}+b_{n}^{2}}\;\;, x\in \mathbb R $ such that $0
When is the function $f(x)$ bounded away from zero? that is $f(x)\geq a>0$, for some $a>0$, for all $x\in \mathbb R$. More specifically, when $\lim_{x\rightarrow -\infty} f(x)$ is not zero. I believe that this would somehow depend on the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
So, first if the limit is 0, does this mean that the limit of each term in the summation is zero?
Thanks
*Note: I've posted this question here, but I didn't get the appropriate answer, so please don't vote for close! *
The problem with answers to the previous post is that I want to know when $f(x)$ is bounded away from zero and not just examples when it will be zero, there are many cases when it will be zero, and this is not what I want.