Proving $e$ is irrational using its reciprocal and the inequality $0<e^{-1}-s_{2n-1}<\frac{1}{(2n)!}$.

Question

Here is a small part of the proof found on Wikipedia:

I am trying to understand the inequality used for the proof of $e$ being irrational. The inequality is

$0<e^{-1}-s_{2n-1}<\frac{1}{(2n)!}$

Is the middle part the error of the series for $e^{-1}$ after the $n$th term? Or is it for another term in the series? And what does $\frac{1}{(2n)!}$ siginify? Does it have something to do with the neglected terms after the series is stopped?

Answer 1

This results from Leibniz' criterion for alternating series: the error for a convergent alternating series when truncated at the $n$-th term is no more than the next term in absolute value, and it has the same sign.