I'm working with the following conditioned probability density

$\displaystyle \Pr(R|N)= \binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N

$

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

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

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

that gives

$\displaystyle \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 $\displaystyle N$

Is there any other way to find an estimate of $\displaystyle N$given a particular $\displaystyle M$ 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!