The expression $x^n / n!$ appears in a the infinite sum definig $e^x$ and similar terms in the sums defining $\cos(x)$, $\sin(x)$, etc. I would like to know if there is some combinatorial/probabilistic meaning or analogy to the term $x^n / n!$ and appropriately an example of a scenario of selection or decision making that could be represented by a consequtive sum of these terms similar to the infinite sums defining $e^x$ or $\cos(x)/\sin(x)$.
In other words: some probabalistic/Combinatorical scenario whose calculation would converge to one of these functions ($e^x/\cos(x)/\sin(x)$). Thanks alot