Let $(Y_n)$ be i.i.d. random variables taking values $1$ and $-1$ with equal probabilities. I want to compute the function $$ f(x) = \mathbb{P}[\sum_n x_nY_n \text{ converges}] $$ defined on sequences $x = (x_n)$ of real numbers.
Trials:
I am stuck at a early stage. Writing \begin{align*} \mathbb{P}[\sum_n x_n Y_n \text{ converges}] &= 1 - \mathbb{P}[\sum_n x_nY_n\text{ does not converge}], \end{align*} I am trying to show $\mathbb{P}[\sum_n x_n Y_n\text{ does not converge}] = \frac{1}{2}$ if $x_n \not= 0$ and $ = 1$ otherwise. If $x_n = 0$, then obviously the sum converges. If $x_n \not= 0$, then I am trying to use Borel-Cantelli to get some kind of result, but hasn't been making much progress. Am I going in the wrong way here?