# Reference for the rate of divergence of Harmonic series

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?

• Try a Riemann sum for $\int_1^n \frac{dx}{x}$
– πr8
Sep 1, 2016 at 7:01
• IIRC, this is easy to do yourself; repeat wikipedia's "integral test" argument, but aiming to get an upper bound rather than a lower bound.
– user14972
Sep 1, 2016 at 7:03

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 $n\to\infty$.
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.