I am trying to figure out a formula for the unique $\lambda>1$ such that $$ \mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1 $$ where $X$ is a discrete random variable taking values in $\{\frac{1}{n},...,\frac{n-1}{n},1\}$, distributed w.r.t. some distribution $\mu$.
We can rewrite the expression above which yields $$ \sum_{k=1}^n \frac{\mu(\frac{k}{n})\frac{k}{n}}{\lambda-\frac{k}{n}} = 1. $$
I know that there are no closed solutions for the roots of such a function, since they are based on solving for zeros of a high degree polynomial. Still, I think I miss some obvious results here how to analyse such a function.
I'd appreciate any kind of help. Thank's a lot!