## You are here

Homeproof of divergence of harmonic series (by splitting odd and even terms)

## Primary tabs

# proof of divergence of harmonic series (by splitting odd and even terms)

Suppose that the series $\sum_{{n=1}}^{\infty}1/n$ converged. Since all the terms are positive, we could regroup them as we please, in particular, split the series into two series, that of even terms and that of odd terms:

$\sum_{{n=1}}^{{\infty}}{1\over n}=\sum_{{n=1}}^{{\infty}}{1\over 2n}+\sum_{{n=% 1}}^{{\infty}}{1\over 2n-1}$ |

Since $\sum_{{n=1}}^{\infty}1/n=2\sum_{{n=1}}^{\infty}1/(2n)$, we would conclude that

$\sum_{{n=1}}^{{\infty}}{1\over 2n}=\sum_{{n=1}}^{{\infty}}{1\over 2n-1}.$ |

But $2n-1<2n$, hence $1/(2n)<1/(2n-1)$, so we would also have

$\sum_{{n=1}}^{{\infty}}{1\over 2n}<\sum_{{n=1}}^{{\infty}}{1\over 2n-1},$ |

which contradicts the previous conclusion. Thus, the assumption that the series converged is untenable.

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

40A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections