If $X\sim f_X(x\mid\theta)=\frac{1}{2\theta}e^{-|x|/\theta}$, $x\in\mathbb{R}$, and $\theta>0,$ find an unbiased estimator of $(1+\theta)^{-1}$ of least variation.
I have tried using the fact that $|X|\sim \exp(1/\theta)$ and so $T=\sum_{i=1}^n |X_i| \sim \operatorname{gamma}(n,1/\theta)$ . I calculate that $E[1/T]=\frac{1}{(n-1)\theta}$ but I can't calculate something like $E[1/(1+T)]$.