Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form $$\mathop{\sum}\limits_{i=0}^{\infty}k_it^i.$$
I would like to ask:
Is there a general result that assures that there exist a root of $p$ in $(0,1)$ for every $p \in P$? If so, can you provide me a reference?
Same question for $S$: Is there a general result that assures the existence of a zero of the power series in $(0,1)$ for every $s \in S$? I'm excluding the cases where $k_i = -1$ for every $i$ of $k_i = 1$ for $i$. Also, the cases when there exist an $n \in \mathbb{N}$ such that $k_i = 1$ of $k_i = -1$ for every $i \geq n$ are excluded.
If the result for the series exist. Is it possible to claim that the number of ceros of $s$ in $(0,1)$ is finite?
I do feel that this is a realy basic question. Sorry about that. Thanks for your help!
In many cases there is no root in $(0,1)$. For example, assume that $\sum_{i=0}^m k_i>0$ for $0\le m\le n$. Then one quickly shows that $P(x)\ge 1$ for $x\in[0,1]$ because you can pair off each monomial having a negative coefficient with a monomial of lower degree having a positive coefficient.
The same generalizes to the case of series. If $\sum_{i=0}^m k_i>0$ for all $m\in\mathbb N_0$, then for all $x\in[0,1]$ all partial sums are $\ge 1$, hence the limit (if it exists) is also $\ge 1$. One example of this type is $f(x)=\frac{1+x-2x^2}{1-x^2}=1+x-x^2+x^3-x^4+x^5-x^6\pm\ldots$