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:

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

The terms in red is smaller than the corresponding term of H. Let's group the like terms together:

H1+12+(\textcolorred14+14)+(\textcolorred16+16)+(\textcolorred18+18)+(\textcolorred110+110)+...

Adding the like terms:

H1+12+(\textcolordimgray12)+(\textcolordimgray13)+(\textcolordimgray14)+(\textcolordimgray15)+...

Simplify:

H1+12+\textcolordimgrayH1H12+\textcolordimgrayH

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

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