Equation with expected value of inverse of a binomial variable I need to solve an equation $\Bbb E\frac{1}{a + X} = b$ where $X\sim \mathrm{Bin}(p; n)$ in terms of $p$, where $a,b&gt; 0$ and integer $n$ are given. As far as I understood from this question, the LHS does not admit a particularly elegant closed form, so no big luck inverting it. On the other hand, I rather know an approximate value of $p$ for $n$ large enough. Perhaps, in that case it may make sense to assume that $X$ is almost normal: I have not found any results on how hard is to compute the LHS in that case. Any ideas?
If you don't want to use the LHS closed-form from the answer in the your link, you can in that closed form substitute $Q=(1-p)^n \, _2F_1\left(a,-n;a+1;p/(p-1)\right)$ by some approximation. To make one note that

$$Q^{(m)}(0)=\frac{(m!)^2 \binom{n}{m}}{\underset{i=0}{\overset{m}{\prod }}(a+i)}\quad,\quad Q^{(m)}(1)=\frac{(m!)^2 \binom{n}{m}}{\underset{i=n-m}{\overset{n}{\prod}}(a+i)}$$

It doesn't make sense to consider just $n$ large, because the LHS converge to 0 pointwise except for $p=0$. (I suppose you want $0\leq p\leq1$)
When $p$ is constant, we have that $E\left[\frac{1}{1+X}\right]=O\left(\frac{1}{n\cdot p}\right)$
as $n\rightarrow\infty$ as shown by Cribari-Neto et al. (2000). See also the paper by Chao, M. T., and W. E. Strawderman. “Negative Moments of Positive Random Variables.” Journal of the American Statistical Association, vol. 67, no. 338, 1972, pp. 429–431. JSTOR, www.jstor.org/stable/2284399.