# How does one show that there is no unbiased estimator of λ−1\lambda^{-1} for a Poisson distribution with mean λ\lambda?

Suppose that $X_{0},X_{1},\ldots,X_{n}$ are i.i.d. random variables that follow the Poisson distribution with mean $\lambda$. How can I prove that there is no unbiased estimator of the quantity $\dfrac{1}{\lambda}$?

Assume that $g(X_0, \ldots, X_n)$ is an unbiased estimator of $1/\lambda$, that is,
Then multiplying by $\lambda e^{(n + 1) \lambda}$ and invoking the MacLaurin series of $e^{(n + 1) \lambda}$ we can write the equality as