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

Failed to parse (syntax error): {\displaystyle {{}} \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Harmonic series/Example]] __NOINDEX__ 2^n+1}^{ 2^{n+1} } \frac{1}{k} \geq \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Harmonic series/Example]] __NOINDEX__ 2^n+1}^{ 2^{n+1} } \frac{1}{2^{n+1} } = 2^n \frac{1}{2^{n+1} } = \frac{1}{2} \, . }

Therefore,

Failed to parse (syntax error): {\displaystyle {{}} \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Harmonic series/Example]] __NOINDEX__ 1}^{ 2^{n+1} } \frac{1}{k} = 1+ \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Harmonic series/Example]] __NOINDEX__ 0}^n \left( \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Harmonic series/Example]] __NOINDEX__ 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

not convergent.