How good an approximation is sampling with replacement to sampling without replacement? I'm learning about probability with Feller's book and he states that, when the population size $n$ is big in comparison with the sample size $r$, then $n_r$, which is a shorthand for $\frac{ n!}{(n-r)!}$, is very close to $n^r$. This is very clear.
What I was curious about was to find a heuristic to how big $n/r$ should be in order to have a good approximation. I reached the conclusion that
$$ |n_r / n^r - 1| < r^2/n 
$$
, when $ r^2 / n $  is small.
The question is if this heuristic is well known and if there is a better simple bound on the above difference.
 A: My answer largely relates to the second part but I may come back with a few words on the first part later (once the notation in the question is clarified)
A common rule of thumb when approximating the hypergeometric by the binomial (which is at heart what we're talking about) is that the sample should be smaller than 10% of the population (another common one says the sample fraction should be no more than 5%).
We can work out the degree of approximation that involves for large samples (large enough for a normal approximation to hold in both cases) by comparing standard deviations (we can then get implied ratios of densities or upper tail areas or differences in cdfs or whatever is desired at some given value of the argument from that).
The ratio of the standard deviations is directly incorporated into the finite population correction factor:
$\text{FPC} =\sqrt{\frac{N-n}{N-1}}$, which in your notation is $\sqrt{\frac{n-r}{n-1}}=\sqrt{1-\frac{r-1}{n-1}}\approx \sqrt{1-\frac{r}{n}}=\sqrt{1-f}$ when $n\gg r\gg 1$, where $f$ is the sampling fraction, $r/n$. When $r$ is small compared to $n$ this is approximately $1-\frac{f}{2}$, so requiring $f$ be no more than 10% is the same as tolerating up to a 5% error in the standard deviation; if you want no more than a 1% error in the standard deviation, this suggests that $r$ should be no more than 2% of $n$, and so forth.
Quite a few posts on site discuss the finite population correction.
