The Problem: Suppose that $Z\thicksim\text{Poisson}(\lambda)$. Find the expected value of $\dfrac{1}{1+Z}.$
We have that \begin{align*} E\left[\frac{1}{1+Z}\right]&=\sum_{k=0}^\infty\frac{1}{1+k}\frac{e^{-\lambda}\lambda^k}{k!}=e^{-\lambda}\sum_{k=0}^\infty\frac{\lambda^k}{(k+1)!}=\frac{e^{-\lambda}}{\lambda}\left[\sum_{k=0}^\infty\frac{\lambda^k}{k!}-1\right]\\ &=\frac{e^{-\lambda}}{\lambda}[e^\lambda-1]=\frac{1}{\lambda}-\frac{e^{-\lambda}}{\lambda}, \end{align*} where we used the Taylor series for the exponential function.
Do you agree with my approach above? Any feedback is most welcomed.
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