I want to know what happens to the bias of the MLE estimator of the geometric distribution with support $\{0,1,2,...\}$ as $n \to \infty$. I arrived at
$\hat{p}_{MLE} = \frac{1}{1+\bar{X}}$
I know this estimator is consistent, but I cannot assert that consistency implies that the bias tends to zero as $n \to \infty$. I know it is biased by Jensen's inequality, and that $n\bar{X}$ has a negative binomial distribution. I tried to compute analytically the mean through the Law of the unconscious statistician, but can't solve it. How can I proceed to assess the "asymptotic" bias?
What I tried is to solve
$\sum_{k = 0}^\infty \frac{n}{n+k}C(n+k-1,k) p^n (1-p)^k$