How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?

Question

Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \begin{equation} \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}\,. \end{equation} Roughly, it should be proportional to $\log(m^2)$. Since, we know that the Green function of Laplacian in $\mathbb{R}^2$ diverging logarithmically. I got this sum as the trace of some regularization of Laplacian defined on two sphere of radius $h$. I would be enormously pleased if someone could explain it to me. Many thanks.