An Asymptotic Example

Let $N \rightarrow \infty$, we get

$$\sum_{ j=\tau}^{ N-1}\frac{1}{j}\sim\ln\frac{N}{\tau}.$$

We therefor shearch for $\tau$ such that
$\ln\frac{N}{\tau+1}<1$ and $\ln\frac{N}{\tau}>1$,
which leads to:

$$\tau\sim\frac{N}{e}$$