# Sum of Roots of a Cotangent equation

I want to reproduce a model calculation, which seemed straight forward, but one piece is puzzling. The (quite simplified) function is:

Sum[1/a_n + a_n/(1+a_n^2),{n,1,Infinity}]


The it says: "a_n, (n=1,2,3, ...) is the nth roots of the equation: ba cot(ba) + cd - 1 = 0", with b,c,d being some unimportant values.I don't know how to find these roots. My problem is in part mathematical, as it is my impression, this equations as infinite positive and negative roots. Maybe it is the idea of this equation, that the sum of the positive and negative is somehow a number. Either way, I don't know how to compute this. When I use something like Solve, Roots, Reduce, ... I never get sensible results.

• You can use any of the methods in this thread to get a pile of zeroes of $ba\cos(ba)+(cd-1)\sin(ba)=0$. – J. M. will be back soon Mar 15 at 14:19
• Thanks, I see. Quite many of these solution don't work. One at least seems to work. Do you think using cos and sin makes it better/faster to compute? – Mockup Dungeon Mar 15 at 15:05
• The idea is to have an equation that doesn't have singularities like the one cotangent has. – J. M. will be back soon Mar 15 at 15:16
• I see – this is working very smooth now. But I have to admit, after running this (even before), this entire instruction eludes me, as it does not provide a sensible set of numbers. The result is basically infinite and cannot sensibly plugged into the Sum term – but that is another problem. I might need to ask the author of the model about this. – Mockup Dungeon Mar 15 at 15:35