Originally Posted by
Moo I'm only curious : how do you get this approximation
Since you ask I would have to give you the error term as well.
Consider the function $\displaystyle f(x) = \frac{1}{x}$ on the interval $\displaystyle [1,n]$ where $\displaystyle n\geq 2$ is an integer.
Then, approximating by Riemann sums,
$\displaystyle \sum_{k=1}^{n-1} \frac{1}{k} \geq \int_1^n \frac{dx}{x} \geq \sum_{k=2}^n\frac{1}{k}$
Thus, $\displaystyle H_n - \frac{1}{n} \geq \log n \geq H_n - 1 \implies |H_n - \log n| \leq \frac{1}{n}$.
This is Mine 94th Post!!!