Does this sum converge or diverge? Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal probability converge or diverge? I have a suspicion that it diverges, but I don't know how to prove it.

I don't know if I exactly have a proof, but here's a thought.  The infinite series of reciprocals of the prime numbers diverges.  How likely is it that $ \ x_n \ > \  1 \ $ "often enough" to produce a series with terms that can bring the series to convergence despite that?  That is, can there be a high enough "density" of terms that make the series convergent against the sum of terms that would cause divergence?  Perhaps there is an argument something like comparing $ \ \sum_{n=1}^{\infty} \ \frac{1}{n^{x_n}} \  $ to $ \ \sum_{n=1}^{\infty} \ \frac{1}{p_n} \  $ , which has a lower "density" of terms than, say, the harmonic series.  (I suspect the probability of having a convergent series is essentially zero.)
