Let $X_n$ be iid $\pm1$ with probability $1/2$. What is the distribution of $\sum_{n=1}^\infty \dfrac{\log n}{n}X_n$?
I can conclude the sum converges almost surely since $\sum_n Var(\dfrac{\log n}{n}X_n)<\infty$. Then I tried with the characteristic function but I am getting an infinite product of $\cos(t\dfrac{\log n}{n})$ and I don't know what happens to this product.
The infinite product $$ \prod_{n=1}^\infty \cos\left(t \frac{\log n}{n}\right) $$ converges since $$\cos\left(t \frac{\log n}{n}\right) \approx 1 - \frac{t^2 \log^2 n}{2 n^2}$$ and $\sum_{n} 1/n^2$ converges. But I don't expect there to be a closed form of this product.