I'm working with the following conditioned probability density

 \Pr(R|N)=   \binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N <br />

where L,R,N are discrete variables, L,R,N>0and R<=L.
What I need is to find the maximum likelihood value for N that maximize the probability to have a given R. The approach I've tried is the classical ML estimation, so I set

\frac{\partial\Pr\left( R=R'|N\right)}{\partial N}=0

that gives

\binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N\ln\frac{r}{L} =0<br />

The problem is that now I'm not able to solve this equation for N
Is there any other way to find an estimate of Ngiven a particular M from the first conditioned probability? I need only an estimate so I can consider to make some approximations...but I need a closed form, I cannot resort to numerical methods.

Thanks in advance!