Jun 8, 2021 - 14:34

Problem

pic.twitter.com/LGWgMgA2px

— 級数bot (@infseriesbot) June 8, 2021

Solution

Let $X$ be a Poisson random variable with parameter $n$. The given expression is equal to $\mathbb{P}[X \leq n]$. For large $n$, it is well known that $Poisson(n) \approx Normal(n,n)$. Therefore the limit of the expression is $\frac{1}{2}$ as the probability of a Normal variable being less than or equal to its mean is $\frac{1}{2}$.