Computing $E[1/X]$ for binomial $X$

Question

Let $X$ be a binomial distributed with parameters $n$ and $p=1/k$. Then we have $E[X]=n/k$. But what is $E[\dfrac{1}{X}\mid X>0]$? Is there a nice closed formula getting the exact value or approximation?

Answer 1

I don't know a closed formula for this, but you could simply calculate it as $E[\frac{1}{X}|X>0]=\sum\limits_{i=1}^n i^{-1}\frac{B(i;n,p)}{1-B(0;n,p)}$