Recently I have come across a variety of truly wonderful results that deal with series that look like harmonic series:
All of the series $$\sum_{n = 1}^{\infty} \frac{1}{n^{1+1/n}}, \ \sum_{n = 1}^{\infty} \frac{|\sin(n)|}{n}, \ \sum_{n = 1}^{\infty} \frac{1}{n^{2 - \epsilon + \sin(n)}}, \ \sum_{n = 1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $$ diverge.
The proof for the latter three mostly hinges on the fact that the fractional parts of $\{\sin(n)\}_{n \in \mathbb{N}}$ are equidistributed in the unit interval.
This leads me to ask the following question:
Let $u_i \sim Unif([0,1])$. What is the probability that $$\sum_{n = 1}^{\infty} \frac{1}{n^{1+ u_n}}$$ diverges ?
Similarly, we can ask an analogue question where $u_i$ are drawn from an arbitrary distribution $X$. I do not much knowledge in this area so any helpful comments and directions are welcomed.
The random variable $\displaystyle {1\over n^{1+u_n}}$ has mean value $\int_0^1 {1\over n^{1+u}}\,du\approx {1\over n\log(n)}$. Since $\sum_{n=2}^\infty {1\over n\log(n)}=\infty$ the random sum also diverges to infinity, with probability one. This follows, for example, from Kolmogorov's three series theorem.
In the three series theorem, the random series either converges with probability 1, or diverges with probability 1. And since the summands are positive, your series must diverge to $+\infty$.