(related to Proposition: Divergence of Harmonic Series)
We want to show that the harmonic series is divergent with
\[\sum_{n=1}^\infty \frac 1n=\infty.\]
Consider special partial sums
\[\begin{array}{rcl}s_{2^{k+1}}&=&\sum_{n=1}^{2^{k+1}}\frac 1n\\ &=&1+\frac 12+\sum_{p=1}^k\left(\sum_{n=2^p+1}^{2^{p+1}}\frac 1n\right)\\ &=&1+\frac 12 \\ && + \left(\frac 13+\frac 14\right) \\ && +\left(\frac 15+\frac 16+\frac 17+\frac 18\right) \\ &&+ \left(\frac 17+\frac 18+\frac 19+\frac 1{10}+\frac 1{11}+\frac 1{12}+\frac 1{13}+\frac 1{14}+\frac 1{15}+\frac 1{16}\right) \\ &&\vdots\\ &&+\left(\sum_{n=2^k+1}^{2^{k+1}}\frac 1n\right).\end{array}\]
Because the sums in all brackets are \(\ge \frac 12\), it follows that \[s_{2^{k+1}}\ge 1+\frac k2.\] Because the sequence of the partial sums \((s_{2^{k + 1}})_{k\in\mathbb N}\) is unbounded, the harmonic series is divergent.
