I am currently trying to place an order of precision on the approximation $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) \approx \left(\frac{s}{t}\right)^b$$
This follows because $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) = e^{ \sum_{r=s}^t \log\left(1-\frac{b}{r}\right)} \approx e^{-\sum_{r=s}^t \frac{b}{r}} \approx e^{-b(\log t - \log s)}$$ via the Taylor expansion $$\log(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots$$ and the approximation $$\log t - \log s \approx \sum_{r=s}^t \frac{1}{r}$$
However, I would like to know how accurate the approximation is in terms of $t$, preferably some big O notation estimate.