Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows $$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\left(\frac{\theta}{\mu + \theta}\right)^{\theta}.$$ Does anyone know how to calculate the expectation of $1 / X$ in this kind of definition of negative binomial distribution?

If there is no explicit expression for $\mathrm{E} (1 / X)$, I wonder whether exists some lower bounds for $$\mathrm{E} \left[ \frac{X}{(X + c)^n}\right],$$ where $c$ is some positive constant and $n \in \mathbb{N}$.

Thanks so much!