I was tasked to find how the series shown below: $$\sum_{n=r}^{\infty} \frac{\left( n-r \right)!}{n!}$$
depends on the integer $r$.
I was given the hint that for $r\le 1$, each term would be greater than or equal to the corresponding term of $\sum \frac{1}{n}$, and that I should do a comparison test with this term, which tells me that the given series is divergent for $r\le 1$ (since $\sum \frac{1}{n}$ is known to be divergent). Another part of the problem gives me a hint for $r\ge 2$, but that is irrelevant to my concern which I will state below.
My problem is, I do not know how to prove that for $r\le 1$, each term would be greater than or equal to the corresponding term of $\sum \frac{1}{n}$ (please spare me your anger, I've just returned from a 4-year AWOL). Can I please get at least a hint (I'd be fine without spoonfeeding) on how to prove it?