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Let $x$ be a small real number, say $1/10 . Let $(a_n)$ be a sequence of integers with $\vert a_n\vert \leq 3^n$. How can I find a positive lower bound for $\left\vert\sum_{n=1}^\infty a_n x^n\right\vert$ assuming that the function $\sum_{n=1}^\infty a_n z^n$ has no zeroes in $\mathbf{R}-\{0\}$.

Using the triangle inequality it's easy to find an upper bound for the above sum, but I don't know any method for finding a lower bound.

Would a lower bound like $\vert a_n\vert \geq 2^n$ help?

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    You could set $m=1$ and get $\left|\textstyle\sum_{n=1}^\infty a_nx^n \right| \ge \left|a_1x \right| - \left|\textstyle\sum_{n=2}^\infty a_nx^n \right|$ for example. But it always depends on what exactly you're trying to do...2011-12-06

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