I am a Calculus II student and we have been learning about series, summation, factorials, etc. I thought about this while I was in the shower and I cannot seem to get it out of my head.
What would this ever be equal to?
$\lim_{p\to \infty}\lim_{t\to p^-}\sum_{n=0}^t \frac{1}{(p-n)!}$
For the first real terms of $n$, it is clearly converged on 0. However, as $t$ approaches $p$ from the left side, the denominator begins to approach $0$. This would indicate that we are now adding terms that approach 1. For instance...
When $t=(p-1)$, the term is $=\frac{1}{p-(p-1)}=\frac{1}{p-p+1}=\frac{1}{1}=1$
Does it converge? Does it diverge? What would this series even look like as $n$ approaches $p^-$?
$$\lim_{p\to \infty}\lim_{t\to p^-}\sum_{n=0}^t \frac{1}{(p-n)!} = \lim_{p\to \infty}\sum_{n=0}^p \frac{1}{(p-n)!} = \lim_{p\to \infty}\sum_{n=0}^p \frac{1}{n!} = e$$