I am trying to compute $\mathbb{E}[X\mathrm{log}(X)]$ where $X$ is a Poisson random variable with mean $\lambda$, so $Pr(X=x) = \frac{e^{-\lambda}\lambda^x}{x!}$. My usual approach to computing $\mathbb{E}[f(X)]$ is to take the Taylor series of $f$ and use the moment generating function, which for X is $M_X(t)=e^{\lambda(e^t-1)}$, and then
$\mathbb{E}[f(X)]=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}M_x^{(n)}(0)$
However, the Taylor series of $f(x)=x\mathrm{log}(x)$ only converges if $|x-1|<1$. Otherwise we have to rewrite the series in a form that's just as hard to work with. And trying to compute the expectation directly fails me. Are there other tricks out there for computing expectations for Poisson r.v.'s?
