<s>[INST] Partial sums of harmonic series I've been given the following problem.

Prove that
$$\lim_{N\to\infty}\sum_{i=1}^N\frac1i=\infty.$$

In other words I need to prove that the partial sum of the harmonic series diverges.  I know the integral test works in this case, but does anybody know of any other methods for showing this?
 [/INST] There are many ways to show this.
\begin{align}
& 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots \\[15pt]
= {} & 1 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots \\[8pt]
& \phantom{1} + \frac 1 2 + \frac 1 4 + \frac 1 6 + \cdots \\[15pt]
> {} & \phantom{{} + {}}  \frac 1 2 + \frac 1 4 + \frac 1 6 + \cdots \\[8pt]
& {} + \frac 1 2 + \frac 1 4 + \frac 1 6 + \cdots
\quad (\text{This “}{>}\text{'' is true if the sum is finite.)} \\[15pt]
= {} & 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots
\end{align}
Here's a more frequently seen way:
\begin{align}
& 1 + \left(\frac 1 2\right) + \left(\frac 1 3 + \frac 1 4\right) + \left( \frac 1 5 + \cdots + \frac 1 8 \right) + \left( \frac 1 9 + \cdots + \frac 1 {16} \right) + \cdots \\[10pt]
\ge {} & 1 + \left(\frac 1 2\right) + \left(\frac 1 4 + \frac 1 4\right) + \left( \frac 1 8 + \cdots + \frac 1 8 \right) + \left( \frac 1 {16} + \cdots + \frac 1 {16} \right) + \cdots \\[10pt]
= {} & 1 + \frac 1 2 + \frac 1 2 + \frac 1 2 + \frac 1 2 + \cdots = \infty.  
\end{align}
</s>