Let $x_i$ be a uniformly random real in $(-1,1)$. And let $f(x)$ be a strictly increasing positive and unbounded function.
Let $S_j(f)=x_0/f(0)+x_1/f(1)+x_2/f(2)+\cdots+x_j/f(j)$
Does $S_j(f)$ converge for every $f$?
Is there an $f$ such that with probability 1, the partial sums $S_j$ change sign infinitely many times and $S_j$ converges?
The answer to both questions is no.
By Kolmogorov's three series theorem, $S_j(f)$ will converge almost surely if and only if $\sum_j {1/f(j)^2}<\infty$. This fails, for instance, if $f(j)=1+\sqrt{j}$.
If $S_j(f)$ converges, it can only change sign infinitely many times provided the limit is zero. If that were true, the equation $x_0=-f(0)\ [x_1/f(1)+x_2/f(2)+\cdots]$ would show that the random variable $x_0$ is independent of itself. But this is impossible for a non-constant random variable.