I am working through this Physics paper. The authors perform a summation over modifed Bessel functions $K_{1}$ for small values of the argument, using the asymptotic relation $K_{1}(z) \approx \frac{1}{z}$ for $z << 1$. $$ \sum_{n = -\infty}^{+\infty} \frac{2}{\pi} \frac{b \sinh^{4}(\pi Q \ell_{S} n / 2)}{[2 \sinh^{2}(\pi Q \ell_{S} n) \xi^{+} \xi^{-}]^{1/2}} K_{1}\Big(\sqrt{8b^{2} \sinh^{2}(\pi Q \ell_{S} n) \xi^{+} \xi^{-}}\Big)$$ $$ \approx \frac{1}{8 \pi \xi^{+} \xi^{-}} \sum_{n} \tanh^{2}(\pi Q \ell_{s}n/2)$$ $$ \approx \frac{1}{8 \pi^{2}Q \ell_{s} \xi^{+}\xi^{-}}\log(2 b^{2} \xi^{+}\xi^{-})$$My question is: How is the last step accomplished i.e. getting from $\sum \tanh^{2}$ to $\log$? The authors claim that the summation is performed over only those values of $n$ for which the argument of the modified Bessel function $K_{1} $ is less than 1.
Thank you.