$\displaystyle x = (1-\alpha) \sum_{r=0}^\infty \alpha^r W^r Y$

where $\displaystyle x \in \mathbb{R}^n, 0 < \alpha < 1, W \in \mathbb{R}^{n x n} $ symmetric ,$\displaystyle Y \in \mathbb{R}^n$.

Derive x in the limit $\displaystyle \alpha \rightarrow 0$.

We are just interested in the sign of $\displaystyle x_i$. It is helpful to expand the sum in $\displaystyle x_i$. What happens as $\displaystyle \alpha \rightarrow 0$? Which terms in the sum are non-zero? Which terms dominate as $\displaystyle \alpha \rightarrow 0$?