Does the sum $$\sum_{z \in \mathbb{Z}^3\setminus \{(0,0,0)\}} \left( \frac{1}{|{\bf x} - {\bf z}|^2} - \frac{1}{|{\bf z}|^2} \right)$$
converge pointwise or even uniformly for $\varepsilon < |{\bf x}| < 1-\varepsilon$?
Convergence of Sum over Integer Lattice
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sequences-and-series
integer-lattices
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0Don't think it converges. At a distance $R\gg 1$ there are $\propto R^2$ points contributing, each with $1/(R-x)^2-1/R^2 \approx 2x/R^3$ such that the sum is logarithmically diverging. – 2011-03-01