I want to know the convergence speed of the following: $$\lim_{\|x\|\to\infty}\sum_{z\in\mathbb{Z}^d}\frac{\|x\|^{d-4}}{\|z\|^{d-2}\|x-z\|^{d-2}}=\int_{\mathbb{R}^d}\frac{dt}{\|t\|^{d-2}\|h-t\|^{d-2}}$$ where $\|\cdot\|$ is a norm equivalent to Euclid norm, and $h$ is an any number of $\mathbb{R}^d$ with $\|h\|=1$.
I think $\sum=\int+O(\|x\|^{-d})$ but I dont know the proof.
Please help me, thank you.