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?