I am looking for the approximation of the following function:
$$\rho(a,b)=1-e^{-(a+b)}\sum_{m=1}^{\infty}\left(\sqrt{\frac{b}{a}}\right)^m I_m(2\sqrt{ab})$$ where $I_m(x)$ is the modified Bessel function. Since $I_m(x)$ is again an infinite series, up to how many terms I need to do summation in both the cases above in order to get some better approximation? Is there any rule of thumb?