I ran into the infinite sum $\sum_{n=1}^\infty 2x\arctan(x/n)n\log(1+x^2/n^2)$, where $x$ is a positive real number. Mathematica can't do the sum, but shows that it's very well approximated by $1.923\cdot10^5x^2$ (tested for values of $x$ between $10^{7}$ and $10^{10}$). Anyone got an idea how to evaluate the sum? What's the constant $1.923\cdot10^5\simeq\exp(12.17)$?
