I'm working on a probability problem, and I end up with the following series $$\sum_{\mathbb{X}\in\mathbb{N}^n}\frac{\left(\sum x_j\right)\lambda^{\sum x_j}}{\prod x_j!}$$ to clear this up, $\lambda>0$ is a real constant and $n\in\mathbb{N}$, and $\mathbb{X}=(x_1,\dots,x_n)$ with $x_j\in\mathbb{N}$. So, it's an infinite series over the $n$-dimensional natural tuples, and for each tuple $x_j$ are its elements.
I need to prove uniform convergence basically because I need to derivate it.
Some context
This appears when trying to prove that a Poisson distribution verifies the Cramer-Rao regularity conditions, what you can see in the series is roughly the expectation of the estimator $\frac{\sum x_j}{n}$ of the Poisson distribution parameter $\lambda$.
Indeed, if a random variable is $X\sim P(\lambda)$ and $(X_1,\dots,X_n)$ is a random sample then the likelihood function is $$L(\mathbb{X},\lambda)=\frac{\lambda^{\sum x_j}}{\prod x_j!}e^{-\lambda n}$$ Defined in all natural $n$-tuples. So the expected value for the estimator $\frac{1}{n}\sum x_j$ is $$\sum_{\mathbb{X}\in\mathbb{N}^n}\frac{\sum x_j}{n}\frac{\lambda^{\sum x_j}}{\prod x_j!}e^{-\lambda n}$$ which leads to the series I posted above. The Cramer-Rao regularity conditions require this to be derivable in order to be able to deduce the Cramer-Rao bound.
EDIT: I've already solved the original problem via an alternative method which did not require to analyze the series, but I'll leave the question open as I still consider it an interesting topic.