Reference for the rate of divergence of Harmonic series

Question

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?

Answer 1

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 $n\to\infty$.

Answer 2

Interpreting the sum as a the sum of the area of $n$ rectangles of size $1\times\frac{1}{k}, \ k=1,\dots,n$ you can bound by an integral.