TITLE: Reference for the rate of divergence of Harmonic series
QUESTION [6 upvotes]: In the wikipedia it states that the rate of divergence of Harmonic series is
$\sum_{k=1}^n \frac{1}{k} < \log n +1 $
I have tried to find a reference, other than wikipedia, for this bound  but with no success. Which is a good reference for the above bound?

REPLY [5 votes]: This bound is obtained by approximating the sum by definite integrals (see here). We obtain
$$
\log(n+1)=\int_1^{n+1}\frac1x\mathrm dx\le\sum_{k=1}^n\frac1k\le1+\int_1^n\frac1x\mathrm dx=1+\log n.
$$
This also shows that
$$
\sum_{k=1}^n\frac1k\sim \log n\quad\text{as}\quad n\to\infty,
$$
where $a_n\sim b_n$ denotes the fact that $a_n/b_n\to1$ as $n\to\infty$.