Harmonic series/Example

The harmonic series is the series

$\sum _{k=1}^{\infty }{\frac {1}{k}}.$

So this series is about the "infinite sum“ of the unit fractions

$1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+\ldots .$

This series diverges: For the ${}2^{n}$ numbers ${}k=2^{n}+1,\ldots ,2^{n+1}$, we have

${}\sum _{k=2^{n}+1}^{2^{n+1}}{\frac {1}{k}}\geq \sum _{k=2^{n}+1}^{2^{n+1}}{\frac {1}{2^{n+1}}}=2^{n}{\frac {1}{2^{n+1}}}={\frac {1}{2}}\,.$

Therefore,

${}\sum _{k=1}^{2^{n+1}}{\frac {1}{k}}=1+\sum _{i=0}^{n}\left(\sum _{k=2^{i}+1}^{2^{i+1}}{\frac {1}{k}}\right)\geq 1+(n+1){\frac {1}{2}}\,.$

Hence, the sequence of the partial sums is unbounded, and so, due to fact, not convergent.