Showing That The Harmonic Series Diverges

Suppose the series converges to \(H\):

\[H = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + ...\]

Then:

\[H \geq 1 + \frac{1}{2} + \textcolor{red}{\frac{1}{4}} + \frac{1}{4} + \textcolor{red}{\frac{1}{6}} + \frac{1}{6} + \textcolor{red}{\frac{1}{8}} + \frac{1}{8} + \textcolor{red}{\frac{1}{10}} + \frac{1}{10} + ...\]

H

H \geq 1 + \frac{1}{2} + (\textcolor r e d \frac{1}{4} + \frac{1}{4}) + (\textcolor r e d \frac{1}{6} + \frac{1}{6}) + (\textcolor r e d \frac{1}{8} + \frac{1}{8}) + (\textcolor r e d \frac{1}{10} + \frac{1}{10}) + . . .

Adding the like terms:

H \geq 1 + \frac{1}{2} + (\textcolor d i m g r a y \frac{1}{2}) + (\textcolor d i m g r a y \frac{1}{3}) + (\textcolor d i m g r a y \frac{1}{4}) + (\textcolor d i m g r a y \frac{1}{5}) + . . .

Simplify:

\begin{aligned} H & \geq 1 + \frac{1}{2} + \textcolor d i m g r a y H - 1 \\ H & \geq \frac{1}{2} + \textcolor d i m g r a y H \end{aligned}

This contradiction shows that the harmonic series doesn't converge.