st.statistics - variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $lambda$

Sorry, I gave a moronic answer before. Let me try to give a better one.

There should be no expression for $f(lambda) := sum_{k geq 1} lambda^k/(k^2 k!)$ in elementary functions. If there were, then $g(lambda) = lambda f'(lambda) = sum_{k geq 1} lambda^{k}/(k cdot k!)$ would also be elementary. But $g(lambda)=int_0^{lambda} frac{e^t-1}{t} dt$ and $e^t/t$ is a standard example of a function without an elementary antiderivative.