Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write \begin{equation} u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} \quad \forall x\in B(0,1) \backslash S, \end{equation} where $\alpha$ is a positive real number.
1) Is the series convergent? If so, in what sense? Is the series uniformly convergent?
I think the series is convergent a.e by the integral test. However, I'm not sure about the uniform convergence.
2) I would like to find the weak (first) derivative of $u$, but do I need uniform convergence to differentiate under the summation?
Thank you.