Suppose $X_1,\ldots,X_n \overset{\mathrm{iid}}{\sim} \mathrm{Geometric}(p)$ so that $P(X_i=x) = (1-p)^{x-1}p$ for $x=1,2,3,\ldots$. Let $\theta=\frac{1}{p}$.
Find $(\mathrm{i})$ the MLE of $\theta$, $(\mathrm{ii})$ the MLE of $p$. Show that $(\mathrm{i})$ is unbiased but that $(\mathrm{ii})$ is biased.
I have shown that $\hat{\theta} = \overline{X}$ and $\hat{p} = \frac{1}{\overline{X}}$, and that $\hat{\theta}$ is unbiased, but how can we show that $\hat{p}$ is biased? Calculating $E\left(\frac{1}{\overline{X}}\right)$ doesn't seem obvious.