## parameter estimation

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
$

where $L,R,N$ are discrete variables, $L,R,N>0$and $R<=L$.
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
$

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 $N$given 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.